Ask HN: Should we use website log analyzers again?

As I'm thinking about privacy on my own blog I'm wondering, "should we use log file analyzers again?" My primary motivation is not sharing my users information with a third party (even one that claims to take privacy seriously).

I'm a bit old school so I remember the days of using awstats and webalizer to read your apache log files and generate stats about your top visited pages, most popular countries, etc.

Are there more modern equivalents of these tools that run on our own servers and do not share data with third parties? Should we be using log file analysis for our primary statics again, especially on small sites?