Here's an example from my field, which I think is informative because it gives a concrete idea of how this plays out in a public arena.
tl;dr for this wall of text: 1) Authors A describes algorithm, 2) Author B publishes counter-example to show where #1 fails, 3) Authors A say it wasn't wrong, but that the author of #2 'misunderstood', and B should have contacted them first; and in any case here are the missing details, 4) Author B points out that paper #1 should have said those details were missing. 5) Authors C point out that authors A misunderstood many things in their own publications; authors A can't complain about others not contacting them first when they don't do it themselves.
"Canonical Numbering and Constitutional Symmetry" (1977), DOI: 10.1021/ci60010a014 describes an algorithm.
"Erroneous Claims Concerning the Perception of Topological Symmetry" (1978), DOI: 10.1021/ci60014a015 points out examples where the algorithm from the first paper, and from another paper, don't work.
The authors of the first paper followup with "On the Misinterpretation of Our Algorithm for the Perception of Constitutional Symmetry" (1979), DOI: 10.1021/ci60017a012 .
> A recent paper in this journal contained critical comments on two methods for the perception of topological symmetry. Carhart’s claim that our algorithm does not correctly perceive topological symmetry and fails with certain structures is the result of a misinterpretation of our algorithm.
> Unfortunately, the author did not contact us directly to help him clarify his misunderstanding. This failure is unusual and difficult to understand. Thus, it was not until we received the recent issue of this journal that we learned of this misinterpretation.
> In our paper we were particularly aiming at catching the interest of the organic chemist for the problems of uniquely numbering the atoms of a molecule. Therefore, we put particular emphasis on the criteria for determining priorities among atoms to enable the chemist to manually number the atoms of molecules according to our procedure. We restrained from giving all small details of the algorithm to keep the paper concise, working under the assumption that persons interested in the details would contact us directly. It is astonishing that Carhart at the point where we did not fully elaborate on the details works with the premise that we misconceived the problem. Initially one should rather assume that other people, too, understand a problem. Only if explicit errors are found
should one digress from this conviction.
Carhart followed up with a letter to the editor, "Perception of Topological Symmetry" (1979) DOI: 10.1021/ci60017a600 :
> I am delighted to see that my critique’ appearing in this Journal has encouraged C. Jochum and J. Gasteiger to present previously unreported steps in their algorithm for the canonical numbering of chemical graphs. They refer to these steps as “small details”, but in fact they are the very essence of any routine which reliably finds unique numberings for, ...
> However, I did not misunderstand their previous article (unless lack of clairvoyance can be classed as misunderstanding); I simply took it at face value. My critical comments, and the counterexamples I presented, were completely appropriate in the context of that article. In contrast with their latest offering, Jochum and Gasteiger’s previous paper did not present a sound and accurate definition of constitutional symmetry, nor did it indicate in any way that crucial steps had been omitted. I am sympathetic with the problems of describing a complex algorithm in the limited space of a journal article, but if space limits the development of a fundamental concept, it is the responsibility of the author to say so, and to indicate that a reader must obtain additional information before he tries to implement the described procedure.
It ended with a letter from still other people writing another letter to the editor, "Canonical Numbering" (1979), DOI: 10.1021/ci60019a600 :
> We have been following with some interest the controversy
appearing in this Journal regarding canonical numbering and various types of The first article by Jochum and Gasteiger contains a number of incorrect and misleading statements about both their work and the work of those who preceded them. ...
> Jochum and Gasteiger also strongly implied that they had a “simple” algorithm which gave complete partitioning, eliminating the need for a comparison step. Carhart correctly pointed out that this was not the case. Subsequent publication of the details of Jochum and Gasteiger’s indicated that it does contain a comparison step ...
> On a more general level Jochum and Gasteiger complain that Carhart did not contact them “directly to help him clarify his misunderstanding”. Yet it is obvious from the large number of misinterpretations and/or misrepresentations which appear in their work that they made no attempt to clarify their misunderstandings by discussing such matters with the original authors. Publishing last on a particular subject accords one considerable power, power that carries with it the responsibility to treat the preceding work with fairness and objectivity.
tl;dr for this wall of text: 1) Authors A describes algorithm, 2) Author B publishes counter-example to show where #1 fails, 3) Authors A say it wasn't wrong, but that the author of #2 'misunderstood', and B should have contacted them first; and in any case here are the missing details, 4) Author B points out that paper #1 should have said those details were missing. 5) Authors C point out that authors A misunderstood many things in their own publications; authors A can't complain about others not contacting them first when they don't do it themselves.
"Canonical Numbering and Constitutional Symmetry" (1977), DOI: 10.1021/ci60010a014 describes an algorithm.
"Erroneous Claims Concerning the Perception of Topological Symmetry" (1978), DOI: 10.1021/ci60014a015 points out examples where the algorithm from the first paper, and from another paper, don't work.
The authors of the first paper followup with "On the Misinterpretation of Our Algorithm for the Perception of Constitutional Symmetry" (1979), DOI: 10.1021/ci60017a012 .
> A recent paper in this journal contained critical comments on two methods for the perception of topological symmetry. Carhart’s claim that our algorithm does not correctly perceive topological symmetry and fails with certain structures is the result of a misinterpretation of our algorithm.
> Unfortunately, the author did not contact us directly to help him clarify his misunderstanding. This failure is unusual and difficult to understand. Thus, it was not until we received the recent issue of this journal that we learned of this misinterpretation.
> In our paper we were particularly aiming at catching the interest of the organic chemist for the problems of uniquely numbering the atoms of a molecule. Therefore, we put particular emphasis on the criteria for determining priorities among atoms to enable the chemist to manually number the atoms of molecules according to our procedure. We restrained from giving all small details of the algorithm to keep the paper concise, working under the assumption that persons interested in the details would contact us directly. It is astonishing that Carhart at the point where we did not fully elaborate on the details works with the premise that we misconceived the problem. Initially one should rather assume that other people, too, understand a problem. Only if explicit errors are found should one digress from this conviction.
Carhart followed up with a letter to the editor, "Perception of Topological Symmetry" (1979) DOI: 10.1021/ci60017a600 :
> I am delighted to see that my critique’ appearing in this Journal has encouraged C. Jochum and J. Gasteiger to present previously unreported steps in their algorithm for the canonical numbering of chemical graphs. They refer to these steps as “small details”, but in fact they are the very essence of any routine which reliably finds unique numberings for, ...
> However, I did not misunderstand their previous article (unless lack of clairvoyance can be classed as misunderstanding); I simply took it at face value. My critical comments, and the counterexamples I presented, were completely appropriate in the context of that article. In contrast with their latest offering, Jochum and Gasteiger’s previous paper did not present a sound and accurate definition of constitutional symmetry, nor did it indicate in any way that crucial steps had been omitted. I am sympathetic with the problems of describing a complex algorithm in the limited space of a journal article, but if space limits the development of a fundamental concept, it is the responsibility of the author to say so, and to indicate that a reader must obtain additional information before he tries to implement the described procedure.
It ended with a letter from still other people writing another letter to the editor, "Canonical Numbering" (1979), DOI: 10.1021/ci60019a600 :
> We have been following with some interest the controversy appearing in this Journal regarding canonical numbering and various types of The first article by Jochum and Gasteiger contains a number of incorrect and misleading statements about both their work and the work of those who preceded them. ...
> Jochum and Gasteiger also strongly implied that they had a “simple” algorithm which gave complete partitioning, eliminating the need for a comparison step. Carhart correctly pointed out that this was not the case. Subsequent publication of the details of Jochum and Gasteiger’s indicated that it does contain a comparison step ...
> On a more general level Jochum and Gasteiger complain that Carhart did not contact them “directly to help him clarify his misunderstanding”. Yet it is obvious from the large number of misinterpretations and/or misrepresentations which appear in their work that they made no attempt to clarify their misunderstandings by discussing such matters with the original authors. Publishing last on a particular subject accords one considerable power, power that carries with it the responsibility to treat the preceding work with fairness and objectivity.