Somebody who actually operates a trunked system will have to answer that. As far as I know multi-site trunked systems don't have seamless roaming like cellular telephone systems do.
They can't switch users to another channel or site while the user is transmitting. The radio waits until it receives the end code on the voice channel and then goes back to the control channel. If it doesn't hear the control channel then it scans for another one which might be at a neighbor site.
Some types of systems can send subaudible channel data over the voice channels but as far as I know that is only for priority scanning and not site roaming.
I manage a trunked system, and I can answer that question....
Real radios do not have two receivers. Nor do scanners.
When a radio is receiving a talkgroup call, it is monitoring low-speed subaudible data on the voice channel. This low-speed data gives information such as active priority talkgroups. If RF channel switching is to take place it is handled by the controller when making a channel grant, after the current transmission ends.
As far as site switching, this is handled between transmissions. As ND5Y noted, if a user's radio is not receiving an adequate signal strength from the control channel, it searches it's list for an active control channel it can hear, then goes through the process of affiliating with the site and registering with the controller; there might be a lag of a few milliseconds while switching sites before the radio receives a new channel grant, but this is generally unnoticeable to the user.
However, when a radio is transmitting, it's pretty much lost to the world until it starts receiving again.