I found the solution to this: Adding the folder containing the .swf files to the Trusted Sources list inside Flash's Global Settings. Wasn't documented anywhere in relation to the security settings that effect LocalConnection so it wasn't an obvious fix. Once I did this, the files communicated fine.
The follow-up question is why this is required when one is inside CEF/OBS, and not when they're both in Standalone players, but I'll take the fact that it works for now. :)