So there are two ways to approach this question (and with the caveat that I'm pretty much only looking to the 19th and 20th century development of small arms, not other weaponry).

The first is that not all countries did in fact have their own weapon design. Major powers which had a decently sized domestic firearms industry would, for the most part, try to rely on that. The United States, France, the UK, Germany... They almost exclusively relied on domestic industry for the production of their military armaments, and additionally, relied on domestic designers in most cases.

There are exceptions. For instance, James Lee was British born, but by the time he designed the action which would bear his name in a string of British rifles, he was working in the US, although the gun would of course be built in the UK. However the P14 rifle was built in the US, on contract by the British in WWI, when they needed to expand production beyond domestic capabilities, which in turn led to the Americans adopting the design and building their own as the M1917.

That is really small potatoes though, and these kinds of exceptions don't change the fact that either country had a strong, healthy, domestic firearms industry which they could rely on for the most part. Not all countries could say the same, however, and this manifests itself in two ways. The first is the ability to manufacture, at all, and in this circumstance, there was no country more important in filling that void than Germany and the Mauser rifle. Contract Mausers, built in Germany, were shipped all over the world, arming a great number of countries that lacked the domestic industry to build their own weaponry (see this post on the Ottomans). In some cases, they would quite literally buy that too, German engineers coming over help domestic plants get up and running, such as in Argentina.

In other cases, although they might have had the industry, countries simply knew a good thing when they saw it, which again was often Mausers. Belgium, for instance, adopted the Model 1889 Mauser for their military, but was able to make them domestically at the Fabrique Nationale, formed specifically to make them in Belgium. The Mauser was one of the most popular designs for military arms from the late 19th-century through the end of World War II, and a number of countries fielded variations which although German in design, were domestic in manufacture.

Bringing up Italy in your OP, they are also worth touching on, as they obviously do have a very long tradition of firearm manufacture, so it is not surprising that they would have built their weapons in Turin, but in point of fact, they often did borrow from foreign designs. The Vetterlis they carried in the late 19th century were in fact a Swiss design, and even the Carcano, although mostly domestic, used elements of Mannlicher in its design, a hardly unusual occurrence as this was the case with almost everyone! The cribbing in the American M1903 Springfield actually resulted in a lawsuit which Mauser won!

Which is a good way to segue to the other main approach. Obviously there is an issue of national pride in being self-sufficient in armament. For design you see a lot of licencing or theft, when a design is good, a design is good, after all, but when you are able to built them yourself, it says something about national capabilities. Not only in that nationalistic way though, but also in the sense of self-sufficiency.

A great example for this would be Greece, who used one of the most fascinating pieces of military hardware in the early 1900s, the M1903 Mannlicher–Schönauer. An Austro-Hungarian weapon, it was an excellent rifle, but also a needlessly complicated and expensive design for something intended to be a mass-military arm. As such, while the design became popular for sportsmen, only Greece, for some bizarre reason, decided to use it for its army. They weren't disappointed, per se, as it was a great gun, but they did nevertheless have buyer's remorse since, buying on contract and not making them domestically, a decade after introduction they suddenly found themselves unable to buy more, Austro-Hungary ending contract production due to WWI and their own needed, and uninterested in continuing to arm a neutral party, especially one who might soon end up in the Entente as Greece did in fact do eventually. Most basically, the lesson here is that if you rely on someone else to arm you, it really sucks when you end up at war with them!

So in sum, the answer is a mix of pride, ability, and the need for self-sufficiency. None of that, in the end, precludes that a design might be bad of course. The Carcano is fairly well known for its under-performance after all, and although attempts were made to correct this, it essentially failed. The issues were more attributable to the design of the round than the design of the action, and in the 1930s the M1938 was supposed to transition the Italians to a new, for effective round, with a nice spitzer bullet to replace the earlier rounded version, but with many still armed with the older version, and maintaining manufacture of ammunition for both, it was a logistical nightmare! The result was basically abandoning the newer round. Some discussion of simply rechambering to use the 8mm Mauser of the Germans was done, but this never happened on a mass scale, although it might have eventually alleviated problems, even if it was unlikely to have been pulled off.

It does highlight one more issue, namely why the Italians didn't switch in the end. Obviously, with the M1938, they were attempting to fix problems on their own. And maybe with a few more years of production and replacement, things would have been better, but with the outbreak of war in 1940, that was impossible. The exigencies of wartime necessitate trying to be simple. Many rifle designs see extra doo-hickeys dropped during wartime production, and less is more. To have done a full about face, and begun making Mausers, and a new ammunition type during a war would have been literally impossible, especially while still supporting the older versions. Nor, really, could Germany have stepped in, their own needs paramount, and certainly unable to provide the millions of arms and ammunition Italy would require if they had actually wished to make such a move. Even the rechambering to 8mm Mauser, which was only done with a small handful of Carcanos, would have been incredibly difficult to pull off if done on a wider scale, with its accompanying logistical problems, and still a great deal of manufacturing problems even if on a lesser scale than new production.