They weren't under any obligation to do so. Privateers were not citizens. The letters granted were in support of the declaration of war. You hire folks when you don't have enough. In this case, the Congress hired ships and crews, not citizens. The Bill of Rights is a completely different issue, it limited the power of the Federal government over its citizenry. Two separate and distinct issues.
Not true. Many privateers were owned by citizens and manned by citizens. http://www.nps.gov/revwar/about_the_revolution/privateers.html
The Congress *hired* no one (in this context). They issued permision for private individuals (and groups of them) to attack shipping of a particular nation or group of nations. The privateers could take and sell any ships or cargo they captured. It was in effect legalized piracy. The "pay" came only in the form of booty. Most of the ships were owned by citizens, and mostly crewed by them as well. But the key point here is ownership. The ships, and the cannon they mounted, were not government owned. The Constitution clearly contemplates that citizens (and others) were to continue owning armed ships, or at least armable ships and the guns to arm them with. (The guns might be stored ashore until needed, to allow more "payload" during times of "peace".)
It wouldn't make much sense for the privateers to only be able to aquire and own their guns after being granted the Letter of Marque, since if they could procure the guns quickly, so could an expanded Navy.