You can implement this feature both with BlazeDS and LiveCycle Data Services. BlazeDS provides remote and messaging features (the messaging features is the one allowing you to synchronize the data between the clients), LCDS extends BlazeDS adding new features like data management (productivity improvements), PDF generation, EDGE server for dealing with DMZ zones), MDA development, portal integration etc. It also adds some advanced messaging features like message conflation, throttling, reliability.
BlazeDS is free and open source and in my opinion a robust solution, you can use it if you want to synchronize the data between clients. LCDS adds a lot of things, but the LCDS customers should have a large budget.
How does it work? There is no JMS behind for this feature (however BlazeDS can integrate with a JMS provider so you can have one client in broswser and the second one running a SWING application). Instead there are some message queues on the server and a publisher - subscriber graph. In order to push the data from the clients to the server there are several choices, the more advanced are available only on LCDS: HTTP polling, HTTP long polling, HTTP streaming, RTMP sockets (LCDS only). All of them are described in details on Damon blog.
If you want to see some code go and download BlazeDS and take a look on the samples, there are several ones showing the messaging features. Also there a tomcat server is bundled in the download, and the samples are already deployed in it.