Thanks for reporting your issues here, guys! We're working with our engineers to iron this out.
There are two views you can choose between when using Etsy from a mobile browser. You can use the mobile optimized view (which formats the site to suit a smaller screen) and the standard view (which will display the site it's formatted on a desktop).
If you're viewing in mobile optimized format you should see the Sign Out link at the bottom of your page.
If you're using the site in standard view on a mobile device you should still see the Sign Out link at the top right of the page. On the desktop version of the site we have tucked the Sign Out link within the Your Account menu, but on a mobile browser in standard view you should still see the Sign Out link in the top right of the page.
If you are viewing the site on a mobile browser in standard view and do not see this Sign Out link could you send us a bit of information?
* a screenshot of what appears where you should see the sign out link
* what kind of device you are using
* what operating system your device is running on
* what mobile browser you're using.
You can send us this information here: www.etsy.com/help/contact
For help on taking a screenshot see here: www.take-a-screenshot.org/
Put ATTN: LISA in the subject line and we'll take a look directly.