During wars in the 20th century, most importantly WW1 and WW2, did normal infantry soldiers only carry a rifle or did they have a pistol as a backup weapon as well?

There is always room for more detailed posts on other armies or eras, but you may want to look at a post from /u/the_howling_cow That shows who in a standard WWII U.S. Army infantry company was issued a pistol.

Basically, a pistol wasn't standard issue.

There are reports from memoirs of U.S. soldiers and marines scavenging/obtaining pistols (and sometimes submachineguns) outside of regulations, but I don't have any information exploring how common this was.