It's fairly easy to answer the first part of this question, because there are actually a decent number of heraldic shields extant from the 13th-15th centuries (here are a whole bunch of them: http://www.larsdatter.com/painted-shields.htm). The majority of them date from the latter end of this timeframe. Some (called targes) are intended for use on horseback in tournaments rather than for battle (where they would more than likely have not lasted long enough to eventually end up in a museum somewhere.) Others, called pavises, are intended for archers or infantry.
Almost without exception, these extant shields are decorated by stretching a gessoed canvas over a wooden frame, although there are also examples using leather or skin rather than canvas, which might have been a more durable covering (and is in fact what is used for the few extant "war shields").
The process of gessoing confirms that these would have been painted after the canvas was laid, as there are several layers of gesso involved. The process of creating is thus the same as detailed in book 6 of Cennini's Il Libro dell'Arte for preparing panels for painting. The resulting shield would then be painted, more than likely with egg tempera but possibly with oil paints. Since this construction method is consistent in the extant examples, there's no reason to doubt that this was the standard method for preparing a tourney shield. And given this process requires a skill (gessoing) that would be known by painters but likely not by the average layperson, it is indeed quite likely that painting of tourney shields was done by professional artisans. There are even some which are carved or have bas relief on them. Painting or gilding on leather is also a skill mentioned by Cennini, so decoration might have been done by the same group of professionals as for canvas-covered shields.

That still leaves open the question of more ordinary types of shields (rather than the fancy ceremonial types that survive in museums). There's no reason to suspect that these were any different than their fancier cousins in basic construction, since covering a wooden shield with leather or canvas would help keep it from splintering or shattering when hit. Because of the specialist skills involved in preparing both paint and gesso, I am inclined to favour the idea that shields were constructed and painted by professionals.

Carving (other than to provide basic shaping or ribs) is not likely for anything other than a decorative shield--it's just too much work for something designed for functionality first and foremost. Cennini actually mentions in his treatise about how to make helm crests out of something resembling papier mache--another good indication that artisans are doing this work. A "heraldic design shop" is not likely with off-the-shelf motifs that could be simply attached to a shield, given the uniqueness (at least regionally) of each heraldic design. (You might have three people with arms featuring lions, but they might be different sizes and have different postures (or "poses"). Metal that would add weight to the shield without adding functionality (such as in the form of a boss) would also not be likely.
So to sum up: Fancy shields were definitely painted by professionals. There is a high likelihood that because of the need to know how to make paint, shields made for warfare were also painted by professionals.