"8.8.8.8" might be obtained by DHCP or another protocol from your router or ISP.
There are two kinds of nameservers: authoritative and caching/recursive. Authoritative nameservers tell everyone about a given domain and need to work mostly all the time (there are usually two), no need to have one unless you own a domain. Caching nameservers tell a local network/computer about any domain, they don't need to work when no query is done, i.e. if you have a single computer only, run the cache on it. You already run servers like an X server or cron. The privacy argument that I wrote before is not true in the typical use of DNS: a query is followed by an HTTP or TLS connection which sends the host name in cleartext. I don't know how useful DNS encryption is in practice.
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