The .357 Magnum and the .44 Magnum share virtually identical histories.
The .357 started with the .38 Colt, a stubby cartridge loaded with a heeled bullet. The .44 Mag started with the .44 Smith and Wesson, a stubby cartridge loaded with a heeled bullet.
Both cartridges then went to inside lubricated bullets, bullets with lube grooves that were designed to be seated below the case mouth. Because the new bullets went inside the case, and had no heel, this resulted in a caliber reduction (to around .357 for the .38 Colt and around .429 for the .44 S&W.)
Since the inside lubricated bullet, being seated deeper in the case took up too much powder space, both cartridge cases were lengthened -- producing the .38 Long Colt and the .44 Russian, respectively.
Smith and Wesson then lengthened both cases again, producing the .38 Special and the .44 Special. And finally S&W lengthened both cases again and loaded them to much higher pressure, producing the .357 Magnum in the mid-30s and the .44 Magnum in the mid-50s.
Therefore the .357 Magnum can fire the original .38 Colt and any of its descendants, and the .44 Magnum can fire the .44 American and any of its descendants.