Are you predrillling holes? 2x2 is easy to split. Also, 8' is a long distance for 2x2s to span, without reinforcement you will likely have sloppy corners after moving it just a couple times.
To answer your question 2x2s are 1.5" actual (or close enough to not matter for this purpose). You want the threads to penetrate 3/4 of the way into the second 2x2 if you can manage it - but that length likely isn't available depending on the scrw you use. Recommend #8 x 2 1/2" deck screws (the 3" are just a tad too long). or #8 x 2 3/4" wood screws.
Don't get the "trim" screws, the head is too small, its not good for loads, you can accidentally pull it right through the wood after a year or two. Also, Simpson Strong drive - best avoided. GREAT screws, **IF** you are drilling into a 4x4 - I've had problems with them splitting anything much smaller though.