For one, you can pass a Debug => 1 to new() - debug messages will then be written to STDERR, which you can capture in whichever fashion your shell provides.

For another, if you transfer files using the get() method, then you can store this method's return value which will be the filename the requested file got stored as locally, and then stat this file, as per erasei's suggestion.

And finally, if you use the retr() method and do the transfer logistics manually, you can call bytes_read() on the Net::FTP::dataconn object it returns at any point to query the amount of data transferred so far.

All of this is mentioned in the Net::FTP POD page.

Makeshifts last the longest.