Flickr Wallpaper does not support authentification at the moment. Only your public photos are available through Flickr Wallpaper.
To access your favorites photos from Flickr, you need to make them public.

Click on the Flickr Wallpaper icon in the menu bar. Open the preferences window. In the Flickr tab, select Flickr Most Interesting option.

Click on the Flickr Wallpaper icon in the menu bar. In the menu, choose item Go to photo page. The Flickr page for the current photo being displayed will be open. If you have multiple monitors, a window will appear and you'll be able to pick the photo you are interested in.

Click on the Flickr Wallpaper icon in the menu bar. Open the preferences window. In the General tab, make sure the Use different wallpapers option is checked.

Flickr Wallpaper allows you to customize the photos that get downloaded from Flickr. To do so, click on the Flickr Wallpaper icon in the menu bar. Open the preferences window and go to the Flickr tab. In the Photo Selection section, choose one of the following options:

• Flickr Most Interesting Photos : most interesting photos uploaded to Flickr in the last 24 hours. These are selected by the Flickr team.
• Flickr User Photo Stream : this option allows you to see photos only from a user's public photos. Enter the flickr username in the Flickr username field.
• Flickr User Favourites : this option allows you to see photos only from a user's public favorite photos. Enter the flickr username in the Flickr username field.
• Flickr Group : this option allows you to see photos only from a given Flickr group. Enter the flickr group name in the Group name field. For more information about Flickr Groups, see here
• Tag Search : find pictures based on keywords (also known as tags). Enter the keywords you are interested in the Tags field. For more than one tag, separate them with a comma.