It depended on the country. I can shed some light on the American and Soviet methods of disarmament.

When the Red Army began to demobilize large formations of its troops, the Soviet officials told them to hand over any firearms (government issue or enemy capture) or face potentially being sent to a labor camp. The Soviet government was very careful about how it went about demobilizing its vast army. Its troops had been exposed to capitalist societies, the shortcomings of the government during wartime, and many had witnessed/participated in atrocities that would tarnish the image of the Red Army soldier which was so essential to Russian post-war propaganda. Though the government bestowed numerous gifts upon discharged troops, every soldier who was demobilized also had his/her bag searched on the train before arriving back home. Most did not put up too much resistance to this action because there were still so many weapons and explosives laying about the numerous battlefields back home in Russia. (Source: Ivan's War by Catherine Merridale, pp. 356-357)

In the American military, soldiers were officially allowed to take home one souvenir firearm, and they had to register it before departing back home. The lines to do so were pretty long at places like Camp Lucky Strike in Le Havre, and many soldiers sold off what extra weapons they had. I don't know what the policy was for service weapons or if a service member could send weapons through the mail before being sent to demobilization camps. It wouldn't surprise me if they did.

I can help a bit with Finland. [EDIT: improved sources.]

The Finnish army was disarmed essentially in two phases: the interim peace treaty signed 19 September 1944 stipulated that the army must be demobilized from its peak of over 500 000 troops to peacetime strength of 43 000 in 2.5 months. The latter part, mostly comprised of the youngest troops, still fought in the Lapland War against the retreating Germans. Additionally, the paramilitary Civil Guard had to be dismantled, and with it its own depots and stored weapons.[1]

Planning for demobilization began as soon as the Continuation War began in 1941 [1], as the chaos following the First World War was still within memory and everyone wanted to avoid that. There were fears of widespread unemployment, housing crisis and problems when men returned to their workplaces after years of absence. In principle, they had a legal right to return to their pre-war places of work, but as you can imagine many of those posts had been filled during the war for example.[2]

In practice, Finnish units first moved on foot and by rail to the local area where they had been raised, and demobilized there. At designated demobilization points, the men turned in their weapons and other gear; they could keep their uniforms (with epaulettes cut off) and shoes, but had to turn in hats and belts.[1] (After the war, probably the most common menswear was the old uniform jacket, or something made from it.)

In principle, every weapon had to be turned in, including those captured from the enemy. In practice, quite a few men had already, during the war, smuggled captured trophies back home on leave for example; these trophy guns (mostly pistols) still turn up from old homes and estates. Hiding war trophies was illegal, but many officers seemed to turn a blind eye to that as long as it wasn't overtly conspicuous; on the other hand, I've heard of one sergeant for example who was court martialed, fined severely and demoted to private for hiding captured binoculars.

However, there was also semi-official squirreling away happening: the so-called Weapons Cache Case.[3] Afraid of Soviet designs for Finland, high-ranking officers selected trusted men and began secretly caching equipment for guerrilla war all around Finland. The goal was to gather enough weapons, equipment and supplies for 8000 men, but after careful planning, in a few days enough equipment and food had been cached for about 35 000 men in over 1300 caches all around the country. Most of these weapons were obtained from demobilization depots and had been marked as "damaged beyond repair" or "lost" in official accounts. The operation came to light in spring 1945, and most of the weapons were returned to depots. However, it seems that as the caches were being cleared, at least some of the weapons went missing entirely - taken by men involved, as final "life insurance." There have been cases where a SMG or even a light machine gun turns up during renovation, although these are rare nowadays.

According to Finnish Police, substantial but unknown percentage of illegal firearms in Finland - estimated to be some tens of thousands in total - are thought to be old war trophies, even though thousands have been collected through amnesty legislation. As said, pistols such as Nagant feature prominently.

Sources:

[1] master's thesis of officer cadet Ratinen, J. (2009) Suomen Puolustusvoimien liikekannallepanokyky sodan jälkeisinä vuosina. National Defence College, Helsinki. Particularly pp. 19-23, also pp. 31-34 for Weapons Cache Case.

[2] Holmila, A. and Mikkonen, S. (2015) Suomi sodan jälkeen: Pelon, katkeruuden ja toivon vuodet 1944-1949. Jyväskylä: Atena. Chapter 1 in particular.

[3] Lukkari, M (1992): Asekätkentä (3. täydennetty painos). Helsinki: Otava.