Mulch is supposed to keep moisture in beneath it, but it doesn't require water itself. In other words water your plants when they need it, and let the mulch sit there and do what it does.

All composts are not equal and in order to answer your question with any accuracy we should know what kind it is.

While compost is a great soil amendment I haven't found it to retain moisture nearly as well as shredded bark mulch. I used to mix my limited amount of compost with sawdust/fine wood shavings and the wood kept in the moisture better than straight compost would. However it disappeared quickly and I needed to apply twice a growing season.

True comments. I bought some cheap compost that looks like dirt, because I wanted to use it on the top of my soil where I am working on a lasagna gardening terraforming project (it's not intended to be the top most layer eventually, but it is for now, so I just wanted it to help hold moisture). But because it was mostly dirt-like, it isn't really holding the moisture I expected, even though it is advertised and sold as compost, and I had expected "compost" to do this.

So lesson learned but at the same time, it just means pick and choose a particular compost if you want it to act as a moisture-holding mulch.

Just think of mulch simply as a barrier between air and soil. Protective mulch will be dry up top. On the bottom it will be damp. In between there's insulating sorta-dead-air space.

That's pretty much it and that covers a lot of possibilities.

This post was edited by cold_weather_is_evil on Tue, Apr 29, 14 at 2:46

Mulches help maintain moisture in soil but they do not put any there, so watering after applying mulches is necessary. That can be done by putting a watering system under the mulch or thoroughly wetting the mulch as well as the soil from above.
Compost, wood chips, grass clippings, shredded leaves, straw, hay, any form of organic matter will make a good mulch and will do a very good job of what mulches are supposed to do if enough is put down. Too little and the mulch will not do the job. A minimum of 2 inch thickness is needed and 4 is not too much.