Does a Chromecast device contain any private information?

I believe the chromecast will store information from your router in order to connect. Either way, it's never a bad idea to reset electronics before selling them.

There are two ways to reset it:

1. Hold the button next to the micro usb port for 25 seconds, until the light starts flashing, then unplug usb

2. Go into the menu of the chromecast app and there should be a "factory reset chromecast" option

After you factory reset the Chromecast it is safe to sell. On the settings menu for Chromecast there is a factory reset button. Alternatively, there is a black button on the Chromecast. Press it for 25 seconds while connected to the TV to factory reset.

Source: http://www.tomsguide.com/us/factory-reset-google-chromecast,news-18294.html

one advice is to make the factory reset,

AND check that the device no longer is able to auto login to any of the services that you have used the device for, just to be safe.