See this LiveDocs page on LocalConnection:
The LocalConnection class lets you develop SWF files that can send instructions to each other. LocalConnection objects can communicate only among SWF files that are running on the same client computer, but they can be running in different applications--for example, a SWF file running in a browser and a SWF file running in a projector. (A projector is a SWF file saved in a format that can run as a stand-alone application--that is, the projector doesn't require Flash Player to be installed since it is embedded inside the executable file.)
For every LocalConnection communication, there is a sender SWF file and a listener SWF file. The simplest way to use a LocalConnection object is to allow communication only between LocalConnection objects located in the same domain because you won't have security issues.
Applications served from different domains that need to be able to make LocalConnection calls to each other must be granted cross-domain LocalConnection permissions. To do this, the listener must allow the sender permission by using the LocalConnection.allowDomain() or LocalConnection.allowInsecureDomain() methods.
Adobe does not recommend using the LocalConnection.allowInsecureDomain() method because allowing non-HTTPS documents to access HTTPS documents compromises the security offered by HTTPS. It is best that all Flash SWF files that make LocalConnection calls to HTTPS SWF files are served over HTTPS.