Can Mozilla Thunderbird show sent replies in threaded view?

You need to go to the account settings for the account and select Copies & Folders, then check the box "Place replies in the folder of the message being replied to"

Also enable sorting by thread.

THis will replicate the gmail effect I think.

GMail Conversation View does what you want without having to copy replies into your folders. Sometimes it doesn't properly pull in your replies, and it doesn't start conversations if you initiated them, but it is the most similar to gmail I have found.

It "is" built-in on thunderbird now, if you do the right steps:

There is the Global Inbox (if there isn't one, you have to activate it at View >> Folders >> Unified from this tip)

Go on its properties at "Select the folders to search:"
and click on "Choose...", you will see that your top inbox is checked;
just check the "sent" one, click "ok", and "Update".

Now, go to your top "inbox" that is the global one, activate the threaded column and you have a three with all incoming and outgoing messages!

Obs.: thunderbird version 31.4.0 here.

PS.: follow @hvd comment below to restrict the messages shown to a single account (in case of multiple email accounts)!