I'm assuming you are using SharePoint.
How is it that multiple users are trying to edit the same form? Did you use the Publish feature to publish the form to SharePoint, or are you uploading it directly into a document library? Each user should be filling out their own form, and not opening an existing (other user's) form to enter data.
The only reason that they might be trying to use the same form is if either (a) you didn't publish it; or (b) they are trying to save the form as the exact same name as another user is trying (unlikely).
If you need help on Publishing, let me know. Let me know if you're using InfoPath 2007 or InfoPath 2010.