Probably depends on how you were picking them. If you were using something like DiceWare to pick letters from an English document, then yeah, you would nail the frequency. But if you were picking letters at random in your head, I would be very surprised if you got anything more than a very roughest of distributions. Anything more than "lots of e's, few q's" would surprise me.
Humans rather suck at picking random numbers. We skew towards picking numbers that seem "more random", whatever that means (which would probably work in your favour for picking random letters with an English frequency, though I'm not too sure), but we also avoid "randomly" generating streaks of numbers, because we feel those are "less random".
If you put two teams in a room, one flipping a fair coin (and writing down the results), and the other pretending to flip a coin but just faking the results, it is usually very trivial to pick out which team actually flipped the coin. They are going to have surprisingly long streaks of heads or tails.
I don't have any evidence to necessarily suggest it, but I suspect this anti-streak tendency will tend to be strong enough to interfere with any correct frequencies which may otherwise appear. ("Oh my, this is far too many e's in a row..")
Oh, heh, I can see it that way now. I had intended my comment to say that, since you'd be trying to reach that set of ratios to hide things, you'd probably fail miserably against any competent analysis.