There was a model of the Colt Govt Model (1911A1 to the military) called the Service Model Ace. It was .22 LR cal. I believe it was used prior to WWII by the army. It was designed to have recoil similar to the service pistol. Many, many .22 LR cal pistols were used by the armed services for marksmanship competition. I supposed that could be called training but from at least the 1950's to the current time the services had pistol teams that used the .22 LR cal pistols. Probably a thousand times as many were used for competitive shooting as were used by the OSS. If you find a .22 caliber pistol that had markings that identify it with the military services it will almost certainly be pistol team equipment.
There is a book entitled U. S. Martial .22RF Rifles by Thomas D. Batha. My copy is copyrighted 2000 but may still be available. It pretty well answers your question about what .22 Cal rifles were used and when.