There's another dimension to this for the frontline troops; get taken POW in possession of the other side's weapon or equipment and the presumption will be that you killed one of theirs to aquire it. That brings a possibility that the captor would react more harshly than otherwise.
interesting point - I was told by a man who had served in the LRDG that during an operation in Italy he had to contact a ‘partisan’ who promptly turned him into the local Italian military for a reward ... when the Italians picked him up they told him that they in turn had informed the German authorities ( as required to do) of his capture and the Germans were on the way to pick him up and by the way — was he aware of the recent German order that any allied soldier captured with a ‘long knife’ (the F&S ) was to be considered a spy and shot? He got away as you probably surmised - which is another story.
My father served through Sicily and Italy etc and landed at Pachino armed with a thompson. I wasnt aware of this until I read a publication about this ... but had wondered where the 45acp rounds came from I would occasionally find in his effects. So they were in the system and Canadians armed with thompsons are illustrated in Charles Comforts painting ‘Hitler Line’.
It is also quite correct that ‘losing’ your issued weapon (or any other serialized item) was not a smart idea and could result in a significnt pay deduction (and other punishment as determined) - a thompson smg was (according to my father) an expensive item of equipment relative to a soldiers pay so it wasnt considered a trivial issue. While my father was RQ there were times when ‘fighting patrols’ were the recorded justification for equipment that was the. written off due to being ‘lost in action’ .
He brought a number of pistols back home after serving in NW Europe and occupation and all were registered with the RCMP - p38, several lugers, Enfields ... As a kid I was fascinated with the ‘Broomhandle’ C96 and asked why he didnt t have one. He mentioned he had one in Italy but left it in a desk drawer ‘for the next guy’ - he wasnt impressed with it he said. Several of the pistols he kept werent acquired in Italy but I believe primarily came from the enormous German stores captured in Holland; while they were disarming the Germans there. He said an old friend had encouraged him to ‘pick up a few’ because thats what most guys were doing with the view they would fetch a few dollars when they got home - father said there were many sheds packed to the roofs with pistols. As an aside - in Holland - he said they used the huge stocks of German helmets to create road beds for the vehicles by crushing them into the ground with tanks.
As for use of US weapons by Canadian troops - IIRC there are scenes/footage taken at Ortona where at least one Canadian soldier appears to have a M1 Carbine. And fwiw another officer in my old Regiment who served in France and Holland was quite specific that they much preferred US BAR’s to the BREN and grabbed them whenever they could; I dont want to get into a debate about what was better as I dont know - but he apparently did and members of the company he commanded had a strong preference for the BAR.
Based on what I have in my possession the preference for captured enemy equipment included German binoculars (the big night optics) compasses and the small infantry and artillery periscopes. Other (unusual) items acquired were a few small Olivetti portable typewriters (invaluable in HQ Coy) and a few field radios that permitted listening to non military radio brodcasts ... oh yes - while on occupation acquisitions included a zeiss ikonta and a few instruments (barograph and chronometers) that the kreigsmarine had no further use for. And a whole lot of other miscellany including an escape and evasion kit he had been issued early in the war and from which the gold went ‘missing in action’ — according to a rumour that gold may have been recovered in equal amounts by the members of the platoon he commanded when they landed in Brest
For what its worth those are the details as I know it from what my father, and other who were there, told me.