Yes, detectives generally carry radios at all times, at least where I work. They are not required to monitor the radio during their off times but it seems that they generally do when they're home and awake. Beyond the radio, they all carry either cellphones or pagers and are required to carry them at all times. With the exception of scheduled vacations, they must be contactable (is that a word?!) 24/7.
Again, this is just where I work...but the officers do simply talk to each other over the radio. If the conversation is going to be lengthy, they switch to a private channel so they don't tie up the regular channel. But most of them have cellphones too, so it's easy to just call each other if they want to keep something off the radio. Keep in mind, when I say a private channel, it's just one that dispatch can't hear, scanners can still pick it up unless it's encrypted (we use an 800 mhz system, there are a lot of systems out there in use). In the case of some radios (mostly low band, which is not in use too much anymore) an officer on one side of the county might not be able to hear an officer on the other side of the county, and then they would have to ask dispatch to relay information.
Perks, feel free to PM me if you want me to clarify anything.