A typical Access application designed for sharing traditionally consisted of at least two mdb files, one containing the data, which is put on a shared disk, and the other containing the user interface, which is put on each machine that wants to use it. This was because loading all the forms, code, etc round the network was unnecessarily slow ... but this may not be an issue with modern networking.
If the application hasn't been written with the intention of being multi-user then it will mostly work most of the time, but every now and then something will go wrong, when two people try to do things to the same data at the same time. "Go wrong" will at best be an incomprehensible error message and try again in a minute, at worst a mangled database.
Exposing an mdb file over the internet via a share exposed over the internet doesn't sound too clever to me if you haven't set up a VPN. I wouldn't like to think about the security issues - I'd choose a different approach.
What you probably want to do is put the data into MySQL running on a server somewhere. That doesn't stop the application itself being in Access - an application in Access can have its back end pointed at a MySQL database instead of data tables in an mdb file, thus retaining the existing user interface and development environment. This (mostly) solves the "hooked on Access" issue.