communicate among different sites and share some user information

Hi, I need to communicate among sites. The following is what I want to do.

One doctor is working in several hospitals and he has to share his schedule.

His schedule is something like this. I save his schedule information in one table “wp_myscheduletable”. (All the sites has the same table “wp_myscheduletable”.)

In the morning he will arrange and public his schedule like this in site A:

A site (his user id may be 100)
10:00～
12:00～
14:00～
16:00～

Once he updates his schedule in A site, I need B site can be also updated as this:

B site (his user id may be 500)
10:00～
12:00～
14:00～
16:00～

if one patient in B site reserves the doctor’s schedule of 12:00～,

B site (his user id may be 500)
10:00～
12:00～ reserved
14:00～
16:00～

I need A site can be also updated like this:

A site (his user id may be 100)
10:00～
12:00～ reserved
14:00～
16:00～

How to do this? Is there any API helpful?