Arthur Ogus wrote a book on logarithmic geometry, apparently soon to be published, and there is a preprint version on his webpage. The first chapter is about commutative monoids, and in particular, it has a bit about pushouts (starting on page 12, and you can tell your computer to search for other instances of the word).
General pushouts can be quite pathological, but you can say interesting things if the monoids satisfy some properties such as integrality. For example, if all of the monoids are integral and one of the monoids is a group, then the pushout is integral, and its group completion is the pushout of the group completions. This reduces Reid's example to a calculation with groups.