After scaling, there seem to be gaps between my teeth. Did my teeth separate?

Your teeth have not moved apart. The tartar that was filling the spaces between them has been removed, revealing the space that was already there. In particular, when gums that were swollen from inflammation settle down, the gap between the teeth (a black triangle) can look more noticeable. This is not a side effect of scaling but a mark left by gum disease. However, if the space keeps getting bigger, food often gets caught, or your gums keep receding, gum disease may be progressing, so you should have a periodontal consultation.