How to access Steam chat logs?

Steam does not log your chats, at least not on your local machine, though their new EULA agreement does allow them to review your chat logs, they are stored on their servers or hidden in encrypted files on your desktop as no one has yet discovered them.

It is possible to log your chat via 3rd party support however as it requires some of your steam details, I cannot guarantee it's security and therefore will not post a link to it.

If you have the Steam mobile app, you can access previous messages sent to and from Steam friends in chat.

Once you install the mobile app and log in, select the hamburger menu (on iOS and Android, click the friends tab on Windows Phone), and then press chat.

Select the user you want to chat with.

Chat Logger++ is able to capture and log your Steam chats, without requiring any account info.

The author explains that it is able to access the chats via Steam's standard (and locally accessible) communication methods.

The application does not require administrator rights and its source code is available. I'd say it's a pretty safe solution (except for the privacy issues arising from storing log files of chat messages of course).