Sure they did, private ownership of artillery of their era has been allowed continuously. You can still buy them, no background check to my knowledge.
MC0623 Field Cannon w/Carriage and Implements $10,995.00MC0623 Field Cannon w/Carriage and Implements
The carriage is made of oak and ash; painted the original Civil War color of olive drab. Heavy steel tires are 2 1/4" wide with oak felloes; 34" in diameter with 14 spokes. Tread is 45". The steel axle, wood cases as original is 1 1/4" square with a tapered spindle at each end going into a boxing which is pressed into a wooden hub. Spindle ends have a left and right hand nut that keeps the wheels in place. Diameter of the heavy wood hub with the correct reinforcing steel ring is 8". Muzzle of the barrel to the tail of the carriage in 6'10". Parts such as the trunnion plate, trunnion cap plate, lunette and trail plate, pointing rings at the end of the tail piece and many, many other odd parts are all brass. The accessories include: two trail plates, hand spike, sponge, rammer, correct heavy duty steel worm, linstock properly drilled. Shipped by truck ; freight collect. Additional crating fee of $65.00 required. CANNON FORM REQUIRED. CAN BE DOWNLOADED FROM WEBSITE.
You might want to discuss that point with private and amateur artillerist Henry Knox, father of the American Artillery Branch and was elected Secretary of War by Congress in 1785, and in 1789 was appointed Secretary of War in now-President Washington's new cabinet.
Who owned or kept/maintained the canons?