Only the west side (Sonora side) of the pass is technically open to OVER SNOW VEHICLES (ie: sleds/snowmobiles) at any time from what I've heard. Basically the dividing line in that area is the crest of the Sierra, and it runs south from the parking lot to the top of No Name peak and on to Leavitt Peak. Hence you can't legally sled up to, or around, Leavitt Lake. Or do a multi-night, sled-supported, midwinter ski crossing of the pass to Bridgeport and back, which would be a lot of fun.
The east side (Bridgeport side) is also the training grounds for the Marines and I think they use a lot of the eastside for camping, helidrops and live fire training in the winter so they probably try to discourage traffic in the area without actually closing it off.
Here's a link to a cool map I found of training area on the eastside of the pass. It outlines all of the heli LZ's and the live fire ranges. It's interesting to see the target areas right around Leavitt Lake. It's a big download, but it's a great topo map of the area and interesting to check out:
MWTC RTA MAP
There's more maps and info in the "Range Control" section of this website:
Marine Corps Mountain Warfare Training Center