Sorry to wimp out, but the answer is that you're both right, kinda sorta. I know your question was about the type of cutting board, not what they'd be used for, but we'll see in a moment that these two are related. For a while, a plastic cutting board was considered superior to wood because the grooves cut into a wooden board by the knife harbored bacteria that would infect the next food that was cut. Plastic cutting boards, it was argued, were harder than wood, developed fewer grooves, and were thus less likely to pass along harmful microbes. The only problem is that studies showed that wood cutting boards harbored fewer bacteria, not the other way around.
Wood is water-absorbent, so it dries faster, which means shorter bacterial survival.
Wood contains natural antibiotic agents that retard bacterial growth. Chicken is always cooked, so any bacterial threat there is eliminated, but salad isn't, meaning there's a risk of food poisoning. Hence two boards, one for chicken, the other for anything else. The same scenario applies to beef, pork, liver, or other meat. The thinking is that you want a separate board for non-chicken meats since these foods either don't have to be cooked as thoroughly as chicken or cook completely through at lower temperatures.