I dont know if you mean in the U.S.A, but if you do, It wasnt hard at all. The federal communications act of 1934 guranteed the right of any citizen to recieve any signal, so it was legal, too.
In the 1930's, most if not all police agencies that were using radio were using "one way" radio. This just means they had a radio station, and all the cars had recievers. The cars could not talk back to the station. Have you heard the expression in old movies "Calling all cars"?
These police stations were just above the broadcast band at about 1700kc and were AM. Any home radio that could tune up that high could recieve it, and a lot of older american radios are marked "police" up at that end of the band.
By the mid 1950's I think most agencies were probably using FM two-way radios just like today. In the period you ask about, 1946-48, I would guess the system in use probably varied wildly among different agencies, with some still using one-way AM, some using the old AM but adding FM transmitters so the cars could talk back to the station, and some using FM two way as today.
Heres how the California Highway Patrol did it.
http://www.mbay.net/~wb6nvh/chpradio.htm
http://www.mbay.net/~wb6nvh/CHPcycle.htm
And a little about Motorola from the same guy
http://www.mbay.net/~wb6nvh/Motadata.htm
I think in Canada, before world war two, you had to have a license for a radio reciever. I think right afterward you didn't, but I don't know for sure. Maybe some Canadians will jump in here and tell us.
John