Learn Italian: Ognuno protegge la propria riservatezza e la propria vita privata. - Everyone protects their own privacy and their own private life.

Because the subject of the sentence is the indefinite pronoun ognuno (everyone/each one). When the subject is indefinite or impersonal, Italian requires you to use proprio instead of suo/sua to mean "one's own" or "their own".

Since ognuno means "everyone", why does it take the singular verb protegge?

Ognuno translates literally as "each one". Just like in English where we say "everyone protects" rather than "everyone protect", ognuno always takes a third-person singular verb in Italian.

Does propria agree with the subject (ognuno) or with the things being protected?

It agrees with the things being protected. Just like all Italian possessives, proprio changes its ending to match the gender and number of the noun it describes. Both riservatezza (privacy) and vita (life) are feminine singular, so we use propria for both.

Why do we need the article la before propria?