As documented, the sincedb_path default causes files to be written to $HOME, i.e. the home directory of whatever user that Logstash runs as (typically logstash). It's a reasonable assumption that those files will be writable to Logstash. See below for the code that sets sincedb_path dynamically when the user hasn't set it.
And again, if you increase the logging verbosity Logstash will tell you more about the location of the sincedb files.