How does the catalyst in a polyester resin or epoxy work?
The degree of mixing is very important in "two part" epoxy binder systems for coatings and adhesives.

In most cases the two liquids are quite viscous and the liquids themselves are better described as suspensions or colloidal dispersions rather than solutions. Unlike say two simple aqueous solutions when combined our two prepolymer liquids will not readily homogenize in a reasonable amount of time unless they are mechanically mixed in some way.  Even if the liquids do combine very slowly they will often gel or vitrify by the molecules at the interfaces reacting before the liquids homogenize.

In many epoxy systems we have, as Beerhunter stated above, some oligomer or polymer with epoxy functionality in one part and some liquid with amine (or sometimes hydroxyl) functionality in the other. Every amine needs to "find" an oxirane ring to attack or the curing process simply can't occur. The curing process works by nucleiphillic attack from amines to open epoxied rings.

The presence of leftover unreacted amine in cured epoxy materials often results in a phenomenon known as amine blushing or amine blooming. leftover amines or polyamides migrate to the surface and may in some cases react with air to form carbamates. This can be disastrous for desired materials properties like tack, adhesion (for the epoxy as a substrate), or cohesive strength of the bulk material. 

In summary you can't build a robust three dimensional polymer network without thoroughly mixing because all the reactants can't react. Always mix as thoroughly as possible to get complete conversion.

My experience with polyesters is limited so I will not comment on that.