The FCC requires that RF transmitters behave cooperatively - using the correct parts of the spectrum, not producing interference on other parts. Verifying this behavior is a relatively sophisticated task of electrical engineering, requiring specialized knowledge, expensive calibrated measurement equipment, RF-isolated test facilities, etc. Naturally, the cost burden for this testing falls on the manufacturer, not the taxpayer. So... we do allow "just anyone" to sell radio equipment, as long as they can pay to prove it behaves within the limits of the law. But big players like Motorola are in a much better position to do this than a dude in his basement.
Carriers have paid big bucks for exclusive licenses to useful swaths of the spectrum, and in return they get enforcement to make sure only devices they control can transmit on their spectrum. The FCC would not license a radio capable of transmitting on Verizon spectrum not approved by Verizon.
But this is specific to the radio - the FCC doesn't care about the rest of the phone (or whatever device it is). You can buy a cellular radio chip for your Arduino because it is FCC certified and you can only provide inputs like SMS and audio, not an arbitrary signal to transmit.