I would (and have) kept my cards close to my chest when searching for a new job. I think the employee/manager relationship is a delicate one because on the one hand you're human beings and you develop a relationship as human beings working together. There might be some friendship there. On the other hand, your boss represents the company and will do what's best for the company.
When you're looking for a new job, you're doing what's best for you; what's best for the company is secondary. If you let your employer know you're searching, they'll start prying and trying to make changes to make you happy. That's reasonable, but at this point they may make promises they can't keep. Obviously, if you are unhappy and think they can change things, share those concerns, but if they aren't doing enough for you that you want to look for a new job, then you've sort of given up on them.
When you're looking for a new job, you're doing what's best for you; what's best for the company is secondary. If you let your employer know you're searching, they'll start prying and trying to make changes to make you happy. That's reasonable, but at this point they may make promises they can't keep. Obviously, if you are unhappy and think they can change things, share those concerns, but if they aren't doing enough for you that you want to look for a new job, then you've sort of given up on them.