I'm always impressed by how efficiently they work. The air space around here gets crazy busy, and often after frequency change there isn't a single moment for me to get in. I wait patiently and then all of a sudden they end up calling me first
I always think that's cool, they already know I was handed off and call me proactively
It's not always perfect but it generally works pretty well.
Typical hand off procedure:
Sector 1: Initiates radar handoff to next sector a few miles from the sector boundary or ideally as soon as there are no conflicts between the aircraft and the boundary. Aircraft label starts flashing
Sector 2: Sees aircraft flashing and accepts handoff or requests something to prevent an issue before accepting. Aircraft stops flashing
Sector 1: Sees aircraft stopped flashing and the target has the new sector assigned. "Ships" you, aka gives you a frequency change once safety is assured
Sector 2: Calls you up once you enter his sector if he hasn't heard from you. They know how busy their frequency is so typically won't get upset if you didn't jump in the middle of something and were waiting for an opening.
If someone doesn't get shipped 2 will shout down the hall at Sector 1 or call him on the landline to say "hey - double check so and so". Frequencies get lost on the pilot end or sometimes the controller got busy and didn't ship him yet. Some radars have further automated handoffs too, so once you get within X miles of the boundary it will initiate the handoff automatically (not sure if that is a feature of STARS or ERAM though?). Radars and the underlying automation/processing systems that we consider one big thing are really interesting.
If Sector 2 is on their game and sees a target getting awful close but there hasn't been a handoff because 1 is behind 2 may call 1 and clarify what is going on. 1 can then do a "manual" handoff on the phone and 2 will then take the target. Very much a team atmosphere since if there is an airspace bust both controllers slipped up.