The ticketing creation email monitors themselves cannot handle responses, no. But, if you have your email replies active from the cloud replies monitor in Admin > Organization Settings > Email Replies, whenever someone replies to a ticket notification with the reply token in it, that will get sent back to the email address which is being monitored for replies. This will depend on the email client they are using though as to whether or not that reply-to address gets respected when they attempt this. For instance, the Windows 10 Mail app will not respect that address correctly.
Typically though, for the cloud service, clients will have ticket creation monitors, and the replies monitor, both be looking at the same Inbox. The settings can be made so the monitor will do nothing if a creation monitor comes across a message with a reply in it, and vice versa, so they won't grab and move emails they shouldn't be touching. Essentially a creation monitor can leave a reply there to be picked up by the replies monitor, and only then will that message get moved to the Processed folder.
I hope this helps. Please let me know if you have any additional questions.