Depends upon category of CofA.
Any aircraft, whether on CofA or permit can be owned up to 20 ways.
However, unless it's on a public transport CofA (or a microlight with a type-approved permit), you need to be sole owner to be given initial instruction for a license or rating. If you are sole owner then it doesn't matter what sort of permit or CofA it's got.
Which is all in the AIC mentioned by CessnaEng, which if you're registered with AIS is at
http://www.ais.org.uk/aes/pubs/aip/pdf/aic/4W049.PDF
Also remember that you need the aircraft based at a licensed airfield. So, if for-example you are a member of one of the many well run syndicates at Popham, you'd have to relocate to, say, Thruxton, do the instruction from there, then relocate the aircraft back home - even if it has got a public CofA.
G
N.B. There is a clause you can use, which is that the instructor must be doing it entirely freely and out of the goodness of their heart without receiving any payment whatsoever. I've yet to persuade any instructor to do this for me, but have seen it done - the pilot in question only pointed out after his logbook was signed that he couldn't legally pay for the instruction. I think you'll probably only get away with that approach once per instructor unless you know them extremely well.