Is it themself or themselves?

Well that is wrong actually.
“Your actions speak for themselves” is a much better way of putting it. It is “themselves”.

There’s no such word as “themself.” “Themselves” belongs there.

(“Your,” not “you.”)

“Them” is plural, thus it goes with “selves”, the plural form of the reflexive suffix.

There is such a word as “themself”, but it is not standard. Use “themselves”

It would be interesting to see a working example of the use of the word “themself”.

It is “yourself” “herself”, “himself” or “themselves”, singular or plural. No such word as “themself.” It doesn’t make sense.

That’s what I thought @Skaggfacemutt, that’s why I thought it would be interesting to see a working example of the use of “themself”, something even my spellchecker (despite not being capable of consisting of all the words in the English language) fails to pick up as being correct.

@morphail has the right answer here. There is a word “themself”, but it is non-Standard English and for that reason not widely accepted.

Source: The Oxford Dictionaries.

Each person went by themself.

how can “them,” which is a plural noun, have a “self,” which is a singular noun?

Would you say “himselves,” for instance?

“Each person went by herself or himself,” is awkward. How about: “Each person went alone.”

The question addressed in this article is whether ”“themself”” can be used in some cases instead of ”“themselves”” with reference to a singular ”“they”” or a singular indefinite antecedent.
source:http://canada.justice.gc.ca/eng/dept-min/pub/legis/n39.html