The picket bullet was designed with little bearing surface: essentially a round ball with a pointed nose attached, though some of them had a more-or-less flat base. The bullet was designed so that it could be used with a linen patch - if it had a significant parallel-sided body, the patch could not fold up around the bullet.
The swage was designed to compress the cast bullet, thus closing-up any small voids in the casting and insuring that the bullets were all of the same shape. The swage consisted of a cylindrical body shaped inside to the dimensions of the desired finished bullet, hardened and polished, and a base punch. The swage body was not open at both ends, but sometimes had a small air hole at the nose. The cast bullet was weighed, very lightly oiled, dropped nose-first into the swage body, the base punch inserted and the punch given a number of hard whacks with a hammer. The punch was then removed and the swage body inverted and tapped on its nose with a wooden mallet or stick of hardwood - if all went right, the bullet dropped out of the swage. Sometimes the bullet stuck, which led to adventures in extended vocabulary. Bullets were carefully weighed by the user, and only those of nearly the exact same weight were used for serious target shooting.
Because of the shape of the bullet, and the fact that it had no parallel-sided body, an additional accessory was required in loading: the straight starter. This was an item which looks rather like a short starter, but which had a working end sized and shaped to receive the nose of the bullet and seat it straight and central in the bore of the rifle, in which the false muzzle helped a great deal. Hunting rifles used with the picket bullet were not usually fitted with a false muzzle, but still shot accurately when the straight starter was correctly made and used.
The best contemporary explanation of the rifles, equipment and procedures is likely 'Instructions to Young Marksmen - Improved American Rifle', by John Ratcliffe Chapman, originally written in the mid-1840s, and reprinted in 1976 by Bienfeld Publishing, Inc. It may still be in print, but you should be able to find a copy, which is well worth having.
mhb - Mike