On Linux, you must be root or the other user to get process information for processes running as other users, so prepending sudo is most of what you need. In addition to that, on modern Linux systems, ss is tool to use to do this:
$ sudo ss -lptn 'sport = :80'
State Local Address:Port Peer Address:Port
LISTEN 127.0.0.1:80 : users:(("nginx",pid=125004,fd=12))
LISTEN ::1:80 :::* users:(("nginx",pid=125004,fd=11))
You can also use the same invocation you're currently using, but remember to sudo:
$ sudo netstat -nlp | grep :80
tcp 0 0 0.0.0.0:80 0.0.0.0:* LISTEN 125004/nginx
You can also use lsof:
$ sudo lsof -n -i :80 | grep LISTEN
nginx 125004 nginx 3u IPv4 6645 0t0 TCP 0.0.0.0:80 (LISTEN)