I think it depends on the device. I'd wager that most carrier-branded handset variants are indeed set up to only allow updates pushed by the carrier but I'm not sure that applies to everything. I believe iOS updates can still be initiated by the user. I'm not experienced with iOS on Verizon so I don't know if they managed to screw that up too but as far as I know, Apple has held firm on that front.
I don't use Verizon but friends who do tend to complain that they are often stuck waiting for updates. Even when the unbranded version of a handset has updates available, there is no easy way to install them on a Verizon phone due to locked bootloaders and such. If the VZW variant has different hardware, then even a standard factory image won't do you any good since they will be built for the "everyone else" version.
I really do feel for people whose only choice is Verizon. At least with AT&T you can usually just buy the standard hardware and pop in your SIM to avoid carrier obstructionism. With VZW, they really get you by the short hairs.