The 38 Special specifications from the SAAMI specs (scroll down to the pages labeled 14 and 15 on the bottom,
here) are currently 17,000 psi or CUP (usually not the same magnitude, but they are for this cartridge) Maximum Average Pressure (the maximum average peak pressure value for ten rounds of freshly loaded ammunition) for the standard load, and 20,000 psi or CUP for the +P load. So, the +P pressure is almost 18% higher than the standard load. This is not magnum pressure. The 357 Magnum is over twice the pressure of a standard 38 Special.
The reason for the smaller-than-magnum difference is that, unlike the magnum cartridge, which has a longer case, there is nothing to prevent a +P load from being chambered in a revolver that is not meant for the higher pressure. SAAMI keeps this safe by specifying the same proof pressure range for the standard 38 Special and the 38 Special +P. In other words, even a gun meant only for the standard round is expected to be able to tolerate the +P round. I do not know when this became standard, but it is so in both the current SAAMI standard and the 1993 standard. Unless we can narrow this down, +P loads would be questionable in some guns made before 1993.
The original (1953) Model S&W M&P (later Model 12) Airweight revolvers are an example of something very vulnerable to stretching by +P loads, as they were famous for their aluminum cylinders and frames stretching even with standard 38 Special ammunition, at least in the early days. I know the cylinders were switched to steel in 1954, but I don't know when the frame alloy might have been changed. Perhaps someone with more specific S&W knowledge will be able to say.
As to published loads, I have a method of approximating +P from standard loads, but it is complicated enough that I am not sanguine about posting it. Some powders get spikey (have more and higher pressure swings) when you try to increase them, so it is still best to use pressure measuring equipment to do this. Even then, you have to take some steps to try to see how your gun deals with the published lower-pressure loads and what the change in velocity per grain looks like to get an idea of when it is safe.
Never rely on a chronograph alone to tell you what pressure you are getting. It may be used in conjunction with interior ballistics software to get an estimate, but peak pressure goes up faster than velocity as you increase the powder charge. They are not directly proportional to one another except, perhaps, in a derringer, where the barrel is short enough not to let the peak pressure drop much by the time the bullet clears the muzzle. There are other reasons any such approach can fail. Chronograph calibration, for one. A pressure gun is still the best approach.
) but if you're not following tested data, the results, good or bad are all on you.