I have 2 BC-780's, they're great radios.
First thing I want to ask both of you is this....do you have the proper trunkgroups for the agencies you're listening to?
In order to hear both sides of the conversations you will need to have those talkgroups programmed. As you've found out, without them the only way you can monitor is to take the radio out of the trunk mode and let it scan the system frequencies. It's hit or miss as to what you'll hear, though. Once you have the proper frequencies and the talkgroups programmed then you'll be able to monitor the trunked systems just like non-trunked systems. What you'll see on the screen is this:
While the radio is scanning in the trunk mode the display will show 'ID SCANNING'.
When it stops on a conversation you'll see the alpha tag you assigned to that talkgroup and above the alpha tag you'll see the frequency.
Watch that frequency....as the conversation continues you'll notice the frequency changing each time each user keys his mic, the alpha tag will remain constant until the conversation is over and the scanner resumes scanning. Remember you can scan conventional and trunked systems at the same time. I use one 780 for conventional scanning and the other for trunking even though both are programmed identically.
Go to the Database section of RR.com and you should both be able to find the talkgroups you're looking for.
Good luck and enjoy!