I'm sure they do - but Brady doesn't run off to Twitter to post the news.
It's silly listening to people talk of conspiracy theories of random drug test. Every month, there are 10 players selected for a random drug test. You are dealing with a possible 75 players that are in the selection pool (team, practice squad and reserve list). If it's "random", a player could be selected more than once... and some people may not get selected at all. It's not "everyone gets their turn", but it's "random".
I'm in the trucking biz and in my world, there are DOT random drug tests. And I see first hand what happens when you are dealing with a small selection pool, but you have to meet a certain percentage of random drug tests monthly.