"Brown bugs" is a pretty non-specific description that could apply to a whole host of insect critters :-) Can you be any more detailed as to the type of mulch or better yet, the identity of the bug in question?

Some mulches will be more attractive to insect habitation than others. Extremely pungent wood mulches like cedar or eucalyptus actually deter a lot of insects. And not all 'bugs'/insects found in mulch are going to be harmful. Many are perfectly benign soil and organic matter dwellers and in fact may actually be beneficial residents. What you want to make sure of is that you don't bring in or encourage something like termites. Google up some images to see if that's what your brown bugs resemble.

I've seen roaches in mulch in some gardens.

They ran away very fast. I could not get a look at them, I only saw them leaving, the motion of them. It could have been some kind of roach, because it moved like a roach would very fast. They were not earwigs, and I have never seen them before. They were really creepy, so I want to ID them just so I could avoid them in case something like this begins to happen to me at some point. They were the size of a medium cricket, one may buy in the store to feed snakes and lizards, about an inch in a half. I just wonder if my guest was referring to these bugs when she commented on my mulch. I have had the same mulch for two years, but I would rather spend more and buy bagged mulch, if that can avoid them. I would assume they come in when you buy infected mulch. I don't know really, I am just guessing.

There are some 8,000,000 species of insects identified today with maybe about 8,000 of those the bad guys. Many of the rest are predators of those bad guys or are pollinators (necessary) or are just there and do nothing that requires extermination. There are a lot of insects that help clean up our environment, beneficial insects, necessary for us to continue living here.
Simply having some "bugs" around is not a bad thing, and "bugs" in mulches are normal. The acsence of insects in mulches is a sign of a problem.

Well, around these parts, which admittedly is about 1500 miles away from you, there is a wood roach that likes decomposing wood. I find them around rotting logs, under bark in my firewood, and around the compost pile. I don't know if what you're seeing are roaches, but they are brown, move very fast, and live in the conditions you describe.

They are different from German Cockroaches which are more commonly found as household pests.

And I suppose if one had them, they could possibly have come in with mulch, but they are quite resourceful and mobile, so they can move in from anywhere if the conditions are right. That means that just because you don't have them now, doesn't mean you won't have some in the future, no matter where you get your mulch.

I don't worry about them until I see one in the house. If I don't get it the cat will. No problem.

Here is a link that might be useful: Google Images of wood roaches

Perhaps centipedes. Brown. Move rapidly. Are predators, thus beneficial .

Pictures?

Ah, centipedes, yes, they do move rapidly. Quite a different appearance from cockroaches so it shouldn't be hard to tell which. Or perhaps it's some San Francisco treat that the rest of us aren't aware of. :-D

I think they were centipedes. I could not get a photo, they move too fast. I went back and looked a few times while I could look. I saw a segmented body and many legs. They were not cockroaches for sure.

They are harmless as far as I know, and if they run away that fast, they shouldn't be much of a bother. Unless they run up your leg.

I read they come from the mulch being too thick, but on the other hand, if the mulch is not thick you get weeds. They need a thick layer of mulch. Mine is thin, because I am cheap and lazy with it.

Insects are generally considered a vital part of the composting process, and rarely a harmful element in mulch. Way better than having a slug problem.