On or after October 1, 1971, no person shall operate and no owner shall permit the operation of any snowmobile or all-terrain vehicle unless the owner holds a valid, effective registration awarded by this state or by another state or by the United States, provided such state or district of registration grants substantially similar privileges for snowmobiles or all-terrain vehicles owned by residents of this state and registered under its laws, and unless the identification number set forth in such registration is displayed on such snowmobile or all-terrain vehicle as prescribed in § 14-381, provided every resident of this state shall obtain such registration from this state under the provisions of § 14-381, before such operation shall be lawful. The provisions of this section shall not apply (1) to the operation of a snowmobile or all-terrain vehicle on premises owned or leased by the owner of such snowmobile or all-terrain vehicle or (2) to the operation of a snowmobile in any organized contest as long as such snowmobile is operated in the contest area, provided the owner of such snowmobile holds a valid, effective registration awarded by this state or by another state or the United States.