scala-loging is just a wrapper around SLF4J, so it should really only depend on which SLF4J implementation you are using (Logback, Log4J 2, etc.), and where you want to send your logs to - but not on scala-logging itself.
For exemple, if you use Logback as an SLF4J implementation and Loggly as a log management service, Loggly provides everything you need in their documentation (please note that I’m only using Loggly as an example - I haven’t used their service I don’t know how they compare to their competitors).