I've seen many variations with on-board tools. I believe that tools were almost never painted coming out of the factories, but once the vehicles got to their units several things can happen.
Good maintenance crews, who had plenty of time and the proper facilities would remove most (if not all) tools before applying the unit's camo pattern. However, if time was a factor (or proximity to the front), I'm sure that some crews just painted over those tools that couldn't be quickly removed.
I have seen photographs showing none, some, or all tools painted over. It really depended on the unit, time during the war, and conditions. Your best evidence would be a unit photo if you can find one.
Enjoy your modeling...