I would assume you want secure communications for your log files so just use scp. Something like "scp remote_host_name:/directory/sub_directory/log_file ." <- that's a period or dot, meaning the current directory for the log file destination. Before this would work, you'd need to set up secure shell private keys so you won't have to put in passwords in your scripts. You can edit crontab to send this out periodically under the proper user id.
Next you need to set up an MTA like sendmail. That's complex, but then you can e-mail from scripts using something like:
cat log_file | mail -s "Here is the email subject" firstname.lastname@example.org
the & is used to run in the background. This will put the log file contents in the body of the email. You don't have to pipe to the mailer file into the mailer. This is just one way to do it. I use it for daily reporting on my servers. It works well. The hardest part was setting up sendmail.
Unfortunately no one can set your system up to do this stuff or give you the specific steps. We can help you with specific challenges for your circumstances or set up, but without comprehensive information, no one can solve your problem outright. So I did the best I could by giving you one general method that works for me.