Yes, pictures that are sent or received with Twilio long codes and shortcodes are stored in your account. Twilio will create a Media Instance Resource and store the Media content that was sent or received.
Twilio will keep that media until you issue an HTTP DELETE. For more information on the Media Instance resource please see our Media API Docs.
Twilio bills for storage on a monthly basis. The first 5GB per month is stored free of charge, additional storage is charged at 20¢ per GB per month.