Yes, other people can see your tags. Tags you think are appropriate for a specific copy of a book will actually be applied to any copy of that book in the whole system.

N.C. - just wanted to let you know that tags are for whatever you want to use them for. SO even though everyone can see it and it will be tagged to all of the copies in the system, using the tag in anyway that is helpful to you is perfectly OK.

N.C. you could use an encrypted type of tag indicating the inscription, it doesn't have to be "has child's name inside" it could be "CN-NC" meaning "child's name" with your initials. Doesn't state to others that there's an inscription inside, but YOU will know what it means!