In Greece it's known as a Communistic song and is always used by the Communistic party!
When you say partisans, two kind of people come in my mind: 1) The Red Army, 2) Jugoslavia's Jozip Broz Tito
From 1943-1945, Italy was torn apart by a civil war in which resistance fighters battled against both the occupying German army and the fascist supporters of the puppet Salo regime headed by Mussolini.
In September, 1943, the city of Naples was liberated through a popular rebellion. In the rest of Italy, the resistance groups were, in the beginning, various independent groups organized by political parties like the Communist Party, the Socialist party, the Action Parry (a Republican liberal socialist party), the Christian Democrats, other minor parties, former officers of the Royal Italian army, groups of anarchists, or even apolitical groups just determined to resist the invasion. (This is Italy, after all. To allude to the saying about Jews, three Italians, four opinions.) Later on, the CLN or Committee of National Liberation tried to exert some central control over the various groups. The numbers of these groups swelled with returning Italian soldiers escaping from German internment camps, escaping Allied prisoners of war, and young men trying to avoid deportation to "work" camps in Germany There were also, increasingly, those sickened by the German atrocities against Italian civilians, even former fascists. It's also true that some "bad actors" freed from prison during the chaos, or just seeking the chaos, joined the groups in the mountains. (The latter is heresy as far as the left is concerned, but it's true nonetheless.)
Depending on their political orientation, the groups took certain names. The Communist brigades were known as the Garibaldi Brigrades, the Azione brigades were Giustizia e Liberta, and the Socialist Brigades were the Matteotti Brigades, named after an Italian Socialist politician kidnapped and murdered by the Fascists after he accused them publicly of voter fraud in the elections and of also using violence to get the desired votes.
My area of Italy (both my mother's area of northwest Tuscany and eastern Liguria, and my father's area of Emilia) was, throughout the Mussolini years, very left leaning. Sarzana, in fact, was the last city in Italy to allow the Fascists to enter after the March on Rome. The anarchists were always particularly well represented there. It's not that there weren't people around who belonged to the Fascist party, but they were mostly bureaucrats who joined because of their positions. There were very few of the rabid fascists who could be found in some areas of Toscana or the Romagna, for example, although they had a presence in Massa. I think part of the reason is that La Spezia was industrialized so early, since it was a naval port, and most of the other people were peasants always skeptical of the fascists. We also had a well-established tradition of liberalism from the days of the Risogimento among the members of the intelligentsia and the middle class. My father didn't put up pictures of the Pope in our home. He put up pictures of Garibaldi and Mazzini. Most people, as I think is the case in most countries, just wanted to be left alone to live their lives.
At any rate, in this area there were Garibaldi Brigades and Matteoti Brigades, and also an International Brigade which was started by escaped British prisoners of war including one of our local heros, Major Gordon Lett, who established his base in Rossano, near Zeri, up near the base of the Apennines. There is a book about the International Brigade called Valley in Flames. After the war, the communists, in my opinion, tried to, and to some extent succeeded in obscuring the efforts of the non-communist and socialist members of the movement. They also obscured another unpleasant truth, which is that while most of the partisan activities were focused and important to aid the allies, some of them were abysmally stupid, calling down retaliation upon the surrounding peasants and townspeople just to blow up a couple of Germans. According the the Communists and Socialists, on the other hand, after saving the honor of the Italian peiople, and liberating the country, the resistance was betrayed, and the country turned on them. And so it goes with Italan politics.
Anyway, all of these groups sang Bella Ciao. It was the common resistance song, although apparently that's not well known.
I did look for some videos in English about the partigiani, but I couldn't find any. This video about Rossano is pretty self-explanatory, however. Oh, Spike Lee did make a movie about the resistance in the Garfagnana next door to us. It gets the atrocities right,but the rest is pure Hollywood...in other words totally made up. It's called "Miracle at St.Anna". A very good non-fiction book about the resistance in the Apennines and the atrocities committed on their supporters is called Silence on Monte Sole.