Translation:Everyone's children participated in the holiday.
In Esperanto you are "participating something" which in english is usually said as "participating in something" Thus the party is the direct ocject.
But you can also say "partoprenis en la festo" (probably influenced by english), which case the party is the place where you performed the action rather than the direct object
I would treat Wiktionary entries with caution. Certainly it is wrong here, because "partopreni" can be either transitive or intransitive, according to Plena Ilustrita Vortaro.
You are right, thanks. But surely unless the -n is there for some other reason, like motion towards, which wouldn't really fit with "partopreni", it signifies that the verb is transitive. But I may have misunderstood what you meant.
Do you mean words like "Ĉies", "kies" and "nenies"? If so they are part of the table of correlatives, which consists of groups of words beginning with i, ti, ki, neni, and ĉi. The es ending refers to possession, so "ies" means "somebody's", "ties" means "that person's", "kies" means "whose" (which person's?), "nenies" means "nobody's" and "ĉies" means "everybody's". You can find a complete table of the correlatives at https://en.wikibooks.org/wiki/Esperanto/Appendix/Table_of_correlatives