A 2 stroke diesel is different from a 2 stroke gasoline engine. It doesn't have the same environmental issues because there is no oil in the intake. The only downside is it needs a supercharger to work so it's not practical in small engines. Most(all?) large diesels like in trains and ships are 2 stroke because of the better power/weight ratio.
2-stroke gasoline engines such as dirtbike engines are run with oil in the gas (thus oil in the intake) for economic and spatial reasons (removing or shrinking lots of parts like oil pumps, oil pans and ports.
It doesn't really lubricate like a 4-stroke where oil is pumped up and then scraped back down into the pan. It is still pumped up but it is stored in devices inside the cylinder liner where it is only dispersed as necessary. Each cylinder instead has oil squirters (receptacles) in the cylinder liners and essentially when the pressure under the piston is low enough that the system has determined there is insufficient oil, spring pressure from the oil squirters forces out enough oil until the pressure in the cylinder (under the piston) is enough to overcome the spring pressure and thus the oil squirters stay closed [1].
