I am surprised that no one has referred the OP to this article about the FBI:
http://www.americanrifleman.org/arti...-fbi-handguns/
While the article could have been so much better than it was, it is a nice little summary. The best part about the article was the gallery of photos, which is apparently no longer available when you click on the link.
As to federal agency firearms from the 1920s to the 1950s, the FBI alone had the Colt Police Positive, Detective Special and variants, Colt Official Police, and various S&W K, J and N frame models.
The FBI tended to issue "duty" guns (four-inch barrels) even though they were to be carried concealed. The Bureau back then had a qualification course that had a stage at sixty (60) yards, and a duty weapon was essential.
The FBI pretty much always issued 38 Special revolvers, leaving the .357 Magnum in the armory, to be drawn when needed, although a great many agents found a way to get one checked out to them permanently.
The last revolver the Bureau issued, the Model 13, was chambered for .357 Magnum, but the standard issue ammo was still the 38 Special +P 158 grain LHP load, with the Magnum ammo authorized on a case by case basis.
An agent friend of mine told me of the time he went to qualify with his Model 13, and the range guy, just to mix it up, required each agent to qualify with the ammo matching the caliber marked on the agent's revolver. My friend was not a happy camper after that firearms day, as he was used to the milder 38 Special load.
From 1920 to 1950, FBI issued these, and perhaps others:
Colt Police Positive, 4 inch
Colt Detective Special, 2 inch
Colt Official Police, 4 inch
S&W M&P or Model 10, 4 inch
S&W Registered Magnum, 5 inch
S&W Model 19, 4 inch round butt
S&W Model 10, 2.5 inch round butt
S&W Model 13, 3 inch round butt
Authorized off-duty or back-up changed over the years, but at times included most steel frame 5 or 6 shot Colt or S&W revolvers. S&Ws in this category were Model 36, 49, 60, among others.
While J. Edgar Hoover and Clyde Tolson each had several handguns, most of those were gifts to them personally. Hoover's actual "bureau property issue gun," which he kept as his primary carry gun until his death, was his original 3 inch Colt .32 Pocket Positive, Serial Number 156600, and it was still checked out to him when he died. Clyde Tolson was issued a new 4 inch Model 10, Serial Number D138446, not terribly long before his death. While Hoover's issue Pocket Positive is still at Quantico, Tolson's Model 10 was unceremoniously destroyed in the "ordinary course of business" in 2001.
The Bureau also destroyed well over one hundred Registered Magnums which were still in service more than 50 years after they were first issued (a testament to the durability of the N Frame Magnum), this policy having been put into effect by the Janet Reno/Bill Clinton brain trust.
The Bureau never officially issued Colt .45 caliber 1911s, but they did issue a number of Colt 1911s in 38 Super. Actually, they would have been dressed in "A1" guise with the frame scallops, and all the A1 improvements.
I suspect most other federal agencies issued similar models, even if the agency did not have as colorful a history as the FBI.
Basically, it would be fair to say that the federal issue gun between 1920 and 1950 would have been a 2 inch or 4 inch Colt or S&W medium frame revolver in 38 Special. That answer to your original question would be correct perhaps 98% of the time.
Almost as interesting as what they carried is the topic of how federal agents were trained prior to Jeff Cooper teaching us all how to properly use the one-hand weapon, the pistol. Here is a short film you may find interesting: