Its sounds as if you are not intending to use FileMaker Server? Assuming your answer is yes the issue is that whoever opens the database first will become the de facto server and all file log-ins and requests will go through that person's computer and copy of FileMaker. This is fine for occasional use but from the sound of things this is MUCH more. Say that person's computer is the slowest? What if they are busy doing a report? The performance of all will be impacted. FileMaker is a database and a very disk intensive one at that. So while FileMaker is designed with a direct local drive as the source it will now have to perform it's over over a slower network connection. Your network could be impacted by the almost constant traffic on top of the typical FMP User traffic and of course ad in web and email etc., and you have some more potential issues. Finally you risk data corruption. If the person's network connection is lost for what ever reason the database would be in an unstable state and while at may be able to be repaired it could lead to greater corruption later. The above is a VERY quick summary of some of the issues you should be aware of.
Having said all this you REALLY should go to a stand alone CPU running FileMaker Server, many advantages including live backups throughout the day. If you can't do that at least get a standalone machine running FileMaker and NOTHING else. Give it a good amount of RAM and hard drive able to hold your solution. You're better off with a cheap older used CPU that meets FMP hardware specs than the scenario you've outlined.
I'l close with a STRONG recommendation that you don't do what you've described.
Filemaker is a great program. As I see it, though, the fact that you can only really share a database if you have a computer which is always on and running Filemaker a HUGE negative in my book. I have worked for 3 major corporations and none of them allowed me to set this up. None of them particularly liked the "dedicated computer always on" thing. On the other hand, all 3 of them have urged me to just store my database on the company intranet and have them write a web-based application to provide the user interface. In all 3 cases, sharing a database that lives on a shared drive would have been a great, easy solution. As for my situation now, I'm about to return Filemaker Pro 9 and give my company IT people a call about that intranet thing. It's more expensive, it affords me less control, but my employer allows it. Too bad for me and too bad for Filemaker.