This is an old post, but I recently went thru this eBay API authentication experience in writing a native mobile app following the eBay XML call format myself and have the answer for others who may find this question.
The eBay developers site has documented their auth token flow for APIs, it talks about desktop apps but just read that as the same for native mobile apps as they never updated their documentation. It is, unfortunately, somewhat involved to setup on your developer.ebay.com dashboard but the steps and screenshots are still accurate with the documentation at the time of this answer. For a native mobile app use the "Authorization" type when you get to that field on your dashboard in "Step 1".
eBay API Auth Token Flow
More Detail on eBay Auth Token Flow
What is not so clear in the documentation for native mobile apps, is for "Step 2" you will also need to have the user login manually via web to signin.ebay.com using android.webkit.WebView or such. There is not an auth API from eBay currently to have your own.
Also note that you have to have the accepted/rejected responses from eBay map over to a URL on your server as shown in the documentation screenshots. There is a field in "Step 1" for each of these URLs/parameters, and they must both be via secure https. I found if I used http you would get an error updating your eBay developer profile. There is no option to map the response to a native app directly.
Once you have all this setup correctly, the flow on the app side revolves around the eBay GetSessionID and FetchToken APIs.
Once you have the auth token it's good for 18 months, unless the user revokes it from their My eBay account page. BTW, this all applies to Apple iOS native apps as well.