Howdy
First, a slight clarification. The Police Positives that were chambered for 38 Special are referred to as Police Positive Specials. They have a slightly longer cylinder and frame than the 'regular' Police Positive which was chambered for the slightly shorter 38 S&W and 38 Colt New Police cartridges. If you look at your grandfather's gun it will probably have a marking something like this on the barrel:
Make sure you see the word 'special' there. If it does not say special, it is a Police Positive chambered for the 38 S&W cartridge, which is not the same as the 38 Special.
Here is a bunch of Police Positive Specials, except for the one on the far right, which is a 22 Rimfire Police Positive Target revolver. The gun in the center is chambered for 32-20, the other three are chambered for 38 Special. The oldest gun in this photo is the one on the left, which was made in 1922, the newest is the one next to the 22, it was made in 1952.
All of these guns are fine to shoot with standard 38 Special ammunition, I would not shoot any +p out of them. Yes, the frame of the Police Positive Special is slightly smaller than a K frame S&W, so the gun is a little bit lighter and recoil will be a bit stouter than with a K frame 38. But I would not call it severe.