You may want to try deleting/re-linking (or refreshing) the linked tables in each of the front-ends. Sometimes a table design change causes issues where if one user's frontend doesn't have their tables refreshed with the new design changes (ie. a new field was added/deleted in the table itself), this could most likely be the cause of your problem.
If your data form is not being updated in another frontend (which links to the same backend tables), (and the form is bound, not unbound), you may have an issue with your network itself or the linked tables need to be refreshed/relinked in the frontends due to design changes. You'd issue a me.requery command to requery that form to update data to the table itself which should then be reflected on other computers without the need to exit/reopen the database. Once the table itself is updated, those changes should automatically show for other users if they closed and re-opened the form to the same record.
For example, you may want to test out this scenario:
1. Open a specific record on your data form on 1 computer.
2. Open that same specific record on your data form on a 2nd computer.
3. Update data on the 1st computer (then making sure the table itself is updated).
4. Have the 2nd user close and re-open the form to that same record. If the new data doesn't then show on the 2nd computer when the user closes/re-opens the form, you may want to check something going on with the network itself since this is not normal (and refreshing/relinking the tables didn't work). I'd suspect something is not working optimal in the user's connection to the network drive (ie. this wouldn't necessarily just be an MSAccess issue.)