In the US, there's a big argument as to whether the Second Amendment applies to private individuals or organized militias. In the Revolutionary War, what was the case? Did all of the soldiers bring muskets that they personally owned from home, or did a larger institutions like the Continental Congress or local militias acquire guns for everyone?
There have been a few good posts on the subject of the Second Amendment and militias, like this recent one by u/PartyMoses.
It's now a political question, the meaning and intent of the Second Amendment. However, as far as applying to individuals or militias, the term "well-regulated militia" in 1787 would have been taken to mean trained soldiers acting under the regular chain of command of officers. When a Continental officer said that his force was irregular, he meant untrained, variously equipped, and raw recruits. And later, when Robert E. Lee disparaged "irregulars", he meant bands of armed guerillas doing what they liked.
As for what the soldiers carried in the War for Independence, the answer is, yes. Baron von Steuben ruefully noted that the men he saw at Valley Forge in 1778 were carrying a variety of muskets, carbines, rifles and fowling pieces. Many of soldiers in the rifle companies brought their own rifles, but some were equipped with them after joining up. Many of the militiamen had bought their own muskets, if they were a part of a Committee of Safety. But far more muskets were needed; hunting guns couldn't take a bayonet. The Continental Congress and the states made an effort to make them, but that was pretty ineffectual, really- it was too hard to set up an armory, too hard to staff it and make it produce, and it became a high-value target to the British and often did not last long. Many thousands of muskets were smuggled in, past the blockade, and thousands of bayonets, many thousands of pounds of gunpowder, etc., mostly from the French.