Finding a sports or media celebrity without a tattoo nowadays might be more difficult than one who has been “inked”. But there is clearly confusion about the ownership of copyright and design right and the stars seem aligned for a big dispute, particularly now the US NFL is reported to be seeking indemnities from players wearing tattoos.
In the UK, the IP rights of new designs will, by default, belong to the tattoo artist. Without any “fair use” provision on which to rely, any unauthorised reproduction technically constitutes copyright infringement. There are some limited defences but these do not include “incidental inclusion”, such as where a photograph is taken primarily of the person rather than the tattoo per se, (see The Football Association Premier League Ltd and others v Panini UK Ltd (Court of Appeal)  EWCA Civ 995 which related to images on clothing rather than the body).
I wonder whether, by analogy with trade marks law, a defence of incidental inclusion might be developed. Perhaps this might work on the basis that the fame of the celebrity in question means the tattoo artist knew his work would be reproduced and exploited as a characteristic of the individual concerned, see for instance model Kate Moss’s brand endorsement of Carphone Warehouse. The position would be different if the individual was inked when they were a nobody and later became famous, that is, under English law there could be no consideration by the tattoo artist for subsequent exploitation and reproduction of his work.
Sports and media celebrities, particularly those who seek to exploit their image, should seek an assignment of all IP rights in their tattoos. Tattoo artists should consider carefully what reproduction and exploitation rights they are willing to give up and protect themselves accordingly.