By "distributed websites" you mean multiple webservers for one FQDN? Usually TLS termination would happen higher up the stack than on the webservers themselves (reverse proxy, L7 load balancer, etc) and the cert(s) would live there. But if your infrastructure isn't that complicated then yes, the happy path is have each webserver independently handle its own certificate (but note your issuance rate limits, 5 certs per week for the exact same hostname[1]).

[1] https://letsencrypt.org/docs/rate-limits/#new-certificates-p...