I have had a square grid map for gaming use for over a decade. The squares have always been used to help calculate or estimate distances between things and to measure out rooms / other things as they are drawn onto it.
I voted "sometimes" - but it is really just a matter of convenience. Where the distance to be measured goes basically along the lines, counting squares is the quick way to go. For diagonals or other weird angles, we estimate - and if it is close, then we actually measure. Most of the time such fine distinctions don't matter that much. And when they do, there's always the issue of how would the CHARACTER know the exact distance to such a degree to be standing in just the right spot, just out of (or in?) range of whatever spell effect (for instance). In those cases, we'd always end up making rolls of some sort if it was close.
Though we DID have the common, semi-joke that when the party was moving it was ALWAYS 22 1/2 feet apart (just for safety) so that no two people could be caught in the 20 foot radius of a fireball...