Well, I do hate the neighbors.
Not too many liquid cooled mowers, especially walk behind, so your white smoke isn't from coolant or water. If there was water in the gas it would be hard or impossible to start.
Blue smoke is from oil; flatheads can burn oil from worn piston rings or intake valve stem guide, or theoretically from a plugged crankcase vent system but I've not personally encountered that.
Many mowers are OHV now so bad head gaskets can be added to the list, and some are prone to have that problem. Another symptom of a blown head gasket on an OHV engine is if it madly spews oil out the filler if you open it while it's running (combustion/compression is pressurizing the crankcase). And oil is getting into the combustion chamber so the sparkplug will be filthy.
Also, overfilling the crankcase with oil can cause smoking. For your engine, find out if the dipstick- if there is one- is supposed to be screwed in or merely resting on the inlet for taking a reading.
Black smoke is from a too-rich condition. Too much fuel = bad/maladjusted carb float or fuel inlet needle or just main plus low speed jets maladjusted or restriction in the intake, usually a dirty air filter or choke that isn't opening. Paper air filters that have been gasoline-soaked may appear OK but they don't pass air at all! If I suspect a bad air filter I simply take it off and see if it starts easily with no filter. I've gotten a few engines for free for that reason because the owners gave up on them when they wouldn't start no matter what.