At work, we have a number of long-running scripts. It has been suggested we turn them into modules.

I'd suggest a different approach. Go through your scripts to see what they have in common that can be extracted to a set of modules. Write the modules and then re-write your scripts to use those modules. As you say that the scripts have a "similar structure", it is likely you'll find a lot of code duplication. It's that duplication that you want to get rid of.

For the sake of definiteness, suppose we want to call the module in such a way that it never dies and so that all output emitted by the module is written to a log file and to STDOUT simultaneously.

While you are overhauling your scripts to take advantage of code reuse, be on the lookout for opportunities to reuse code that has already been written. For example, IO::Tee should help you with teeing your output.

-sauoq
"My two cents aren't worth a dime.";