F2Ps can indeed engage in a trade, however, TF2 items remain untradable, meaning they can only accept premium items and trade said premium items.
Truth is, I begged, when I got a few weapons and Scorched Crates (from people who have a ton of them), and I resell them to buyers, and as I slowly gain more money, I can make bigger investments and earn more, I dealt with tools, popular hats/miscs (Familiar Fez, Last Breath, Team Captain...etc) and basically trade 24/7 a day.
GPop got it mostly right, save for one part, my Steam account is upgraded after I bought Portal 2 with a few keys, so I didn't actually buy games, but got games for free. :3