People that trade you can still give you things, and those things can be traded without restrictions.
But should someone give you something usable for crafting, such as a refined metal you can break down into scrap metal pieces, the result won't be tradeable while you're F2P. --- http://i39.tinypic.com/xnar69.gif
f2p players can't trade, or rather they can't give anything, however they can receive things (don't expect people to give you stuff, very few people will do so)
if the items are marked as being unable to be traded, it's because they were likely unlocked by achievements, see here for more info: http://wiki.teamfortress.com/wiki/Achievements#Class_specific_achievements
reaching a milestone for a class will give you an item for that specific class.
for more information on the difference between f2p and premium players, see here: http://www.gamefaqs.com/boards/437678-team-fortress-2/59562788