Yes, the left-side of a grid might be solvable earlier. Maybe you could have entering 1 or 0 for marked row/column instead of every row (like euklid suggested).

Couldn't you mark certain black square with letters and have people enter the clue value for those ones? Generally these puzzle don't have clues in all black squares. This would ensure easier counting and you can choose the right squares by the way the puzzle is designed.
Other possibility would be marking white squares with letters and have them enter which of those letters end up with a bulb. That eliminates counting all together.