Two options: - (1) at the database level, with whats commonly called ETL (Extract Transform and Load). In the Microsoft world you'd use SSIS (which comes as part of MS SQL) to move data about. This would be a common approach within an enterpruise, particularly if you have a lot of control over then environment.
(2) some sort of "service" based approach. Maybe you provide some sort of interface (like a web service) so that one application can call the other. The issue with that is that you need to build it into the application - but you seem to be after a database driven solution (?)
Have a think about what you're trying to do and who should be responsible for that - are you sure it's the database?
Regarding your specific technology - I'm not sure about MySQL as I haven't used it much myself; I don't know of any 'easy' way to have MySQL and Access talk to each other, so yopu may have to write something.
The data your exchanging - how much and how often? How timely (can you have one poll the other every hour, or does it need to be 'real-time')?
You could consider using a (new) third system that brokered communcation between the two databases, so that they could remain ignorant of each other and the need to update.
Is it likely you'll have a third database to update later (or a 4th, etc...?)
can you change the database platform to something that is common across both / all sites, and which has some sort of messaging / updating system built in?