There are multiple approaches to this logistical issue.
First of all, when capturing enemy weapons, the munitions are captured with them as well, to some degree or another. This can range from the handful of stripper clips carried by the surrendering soldier, to a company's supply train, to entire munition depots.
In some cases, through coincidence or de-factor standardization of ammunition, opposing countries share the same munitions.
9mm Parabellum is a particularly ubiquitous cartridge, being used by the British (Sten Gun), Germans (P08, P38, MP40), and Italians (Beretta Modello 1938) in WWII.
Quoting from "The MP38 and MP40 Submachine Guns":
The fact that all the Germans here were armed with Thompsons is worth comment, especially as the Thompson’s .45 ACP ammunition meant that the guns weren’t compatible with the German forces’ own ammunition stocks. To keep their Thompsons working, therefore, the Germans would have had to replenish their ammunition from captured supplies. If they used the British Sten (unlikely, given the comments I will make below), by contrast, the German troops could utilize their own universal 9mm Parabellum ammunition. The Germans would certainly appreciate the Thompson’s rugged reliability, as they would its terminal effects on human opponents at close ranges. It may well have been that the unit captured a supply of Thompsons and ammunition in one lot, thereby making arming the unit homogenously a sensible decision.
Now, it is key to not that not all 9mm Para is created equally, so to speak. The Italian 9M38 loading intended for submachine guns was "hot" ammunition, which is to say rather powerful. It may not have been wise to use in pistols, especially Lugers.
There are more examples: the Soviet PPSh-41 used a 7.62×25mm cartridge identical to the 7.63×25mm cartridge used by the Germans in their C96. The PPSh-41 was captured in large numbers by the Germans, and the commonality in ammunition made supply rather easy.
Second, there is the option of rechambering the weapon.
This involves either modifying the barrel or providing the weapon with a new barrel, chambered for the standard cartridge.
Quoting from "The Mosin-Nagant Rifle":
Some M1891s were rechambered for the 7.92×57mm Mauser cartridge, but most were simply issued with captured 7.62×54mmR cartridges because of the huge quantities that were readily available. M1891s issued to the German Navy are generally marked on their buttplates, the basic marking being the square Imperial crown over a large ‘M’ (Marine).
Now, it is also key to note that the intended goal of the issued weapon plays a role. A captured Mosin issued to a railway guard in the middle of Germany, who merely needs to guard against spies, has a different ammunition consumption than a Mosin used on the front line.
Again, there are plenty of examples of rechambered weapons. French MAT-49s have been found rechambered for the 7.62×25mm cartridge in Vietnam, Lewis guns have been chambered by the Germans, not to mention this Chauchat in 7.92mm.
Sources:
The MP38 and MP40 Submachine Guns, Alejandro de Quesada, p.58
The Broomhandle Mauser, Jonathan Ferguson, p.11
The Mosin-Nagant Rifle, Bill Harriman, p.33
The Lewis Gun, Neil Grant, p.65
Italian Small Arms of the First and Second World Wars, Ralph Riccio, p.196
