Same operating system is not an issue. Mac and windows systems can share the same file no sweat.
I gather these are two individual computer whose only link is via the internet?
One of you will have to be the host. You open the file with filemaker pro on your system and the other can then launch filemaker and use Open Remote... to find and link to your computer--once you have the internet router correctly configured. This is not something I've done personally and you may need to research how to change settings on your internet router.
Essentially, you have to set up your router to forward requests on a given port to your computer (Look up Port Forwarding). Then you need to figure out your router's IP address. Your other user would enter this IP address in order for Filemaker to find your computer and link to the database. Note that most ISP's give you an IP address that will change from time to time unless you pay for a static IP.
For more information, you can search this forum for posts that refer to Port Forwarding, paying special attention to any posts by DavidAnders as he has posted some very helpful links in many of those.
so if we're using a public university network [and have no access to the router] it's kind of impossible??
is there any way to do it with bluetooth?
or create a network from my own laptop?
I'd talk to your campus IT folks to see what they can and will do for you.
It is possible to string a cross over wired ethernet cable between two computers and create a LAN (local area network) of just two computers with no hub or router involved. A friend of mine set things up that way for me when we were volunteering our services and needed two machines to handle volunteer signups at a location where we couldn't get good network access.