You need to explain your setup more, are you running a domain? Are the users using their own usernames or a generic one, how did you setup permissions on the share?
Network share/folder rights should be assigned for the users or groups, not a computer. Your issue would not be coming from the computer but whoever is logged on to it, unless you setup some non-standard rights on the share.
If the share rights are setup for a user, and that user authenticates to a domain, the share assumes the user is valid and grants him/her access to the share without asking for a password again. Meaning is user "Bob" signs on to the domain on any computer, and tries to get to his network share on \\Server1\Share1, and his user or group has rights to that share and folder, that share will open without another password prompt. That's the correct way of setting up shares.
If you just setup rights for a generic account, the user could have checked off "save this password" when connecting.