From a professional perspective, here are two common ways information is exchanged securely and in real time between two or more sites AND avoid proprietary code:
1) An API (advanced programming interface) using XML-RPC
2) XML (such as an RSS newsfeed) <-- I suggest this for non-developers
Proprietary means both sides use a custom solution, non-standardized, which is fine until one side makes a change or needs enhancements.
You could ask the other webmaster to create an API using the XML-RPC method. You (the client) connect and pass request data to them (the server) with XML being used both ways for the transaction. This means you can use whatever software language/framework you want for the client and the webmaster on the other side can use whatever they want as well. RPC stands for remote procedure call, the method by which data is sent between the client and the server. An example API might be eBay which allows you to connect to its server passing your account ID, item ID and other information and it returns data containing all current bidding activity for that auction. All modern web languages have frameworks for setting up either the client or server side API.
XML-RPC tutorial
Here is a PHP framework for XML-RPC (API) for example.
I'd suggest the RSS, of course, as RSS is nothing more than a standardized form of XML. This means, like in the API method, both sides use your own software to process the transaction, it's super easy to implement RSS these days so its the obvious choice for most needs. Or consider an API if you both agree you need customization, dynamic queries of data, and a more "private" direct exchange of data. Unlike RSS, XML-RPC allows for encryption and even license keys if applicable.
-jim