We live in "snow country" but don't tow our trailer during this time of year (it's stored).
If you expect the brakes on your trailer to work effectively you'll need chains or studded tires on the trailer (referred to as "drag chains"
. If you don't use one or the other, the trailer wheels will just lock up on slick surfaces and skid or slide, which may cause other problems. In California, the CHP requires chains on towed trailers, as well as the tow vehicle (snow tires, studded tires, 4x4, etc. may be acceptable for the tow vehicle, depending on highway conditions).
Cable-style "chains" may work well, as opposed to the link-style chains. Make sure you put whatever solution you decide to use on both sides of the trailer, not just one side.
Enjoy your winter travel.