Well, Stretch Tuning is not done to "make them sound in tune with themselves" but one would have to experience a dead-on-the-tempered-scale tuning to understand what's going on there.
The Stretch generates a slight chorusing effect which helps the piano sound, well, like Piano.
But the amount of Stretch is only a few beats, or cents off of the target and the A above middle C should be referenced dead on the 440 in all but the older pianos with pinblock problems.
The amount of pitch difference due to stretch is so minimal that it isn't an intonation problem. If the stretch tune is done properly for the type of piano. There's a different stretch for Grands of different sizes, as well as a different stretch for the Console, for the Upright, for the Upright with Grand harp, and an entirely different stretch for the Spinet. There are even stretch methods for the lowly Rhodes pianos. In the case of any of them, the intent is to get that slight chorusing effect and not really a retune from Tempered Scale at all.
How to tell: A piano that is tuned "dead on" the Tempered Scale, with no stretch, sounds and plays like an "Ice Cream Truck" xylophone. I know this because, when, as a kid still in High School, I saved up my money and got my hands on a Stroboconn and a Piano Tuning toolkit. Let the learning (and the irate Daddy, boy was he mad) begin...

The Digital Stroboconn I use today can store the stretch tables for various pianos in it. Also you can store custom stretch or have it simply memorize a piano with the mic so that you can easily go back and put that particular piano "back to rights" quickly. The piano tuners of the world never had it so good.
--Mac