During a police interrogation you answer questions differently than you do when you are talking to your spouse. For example, “Where have you been?” can be more dangerous coming from one of the two sources.
There are two correct responses to the question, “Where are you interviewing?” If you are talking to a hiring manager or HR person, tell them. Let them know what is going on. Give them details if they ask. It will most likely increase your desirability if they know others are talking to you.
If you are talking to a recruiter at an agency, you need to decide if you trust the recruiter. Ask the recruiter, “Why do you want to know?” After the recruiter acts defensive or offended, ask your real question, “Do you ever submit resumes to jobs you find out about from candidates?”
The recruiter should answer, “I will only submit a resume to a job you mention if I am already working on it, or if you tell me you are out of contention there. I will never reduce your chances of getting a job by submitting competition unless I was already working on the job.”
Do you trust the recruiter? If so, give him the details of your interviews. He can help you much better in your job search if he knows everything. All the recruiting trainers and over half the recruiters will play fair with you. They will not ruin your chances where you are already interviewing. If you have serious doubts about the recruiter, tell them you are interviewing, but not precisely where.
Basically, if someone will hurt you with the information, protect yourself. If the information works to your advantage, tell them.
Evaluate every recruiter you work with. Which ones do you trust? Which ones are questionable? Tell the ones you trust where you are interviewing.