I was wondering if I could compost mesquite and hackberry trees. I would imagine that if I do I would have to chip them up before putting them on the compost pile.

In theory you can hot compost any plant or animal matter.

Smaller portions are best for composting. Since all woody material are "browns", high in cellulose, add lots of high nitrogen matter to the pile, (plus some sugary stuff if you like), to speed up microbial growth and decomposition.