Can non-Garda check who's the registered owner of a vehicle based on number plate?

Is there any way for a non Garda to check who is the registered owner of a vehicle based on the number plate?

yes -- he just asks his garda friend.

Or sends a request to Shannon in writing explaining why.

In cases where legal proceedings are being taken in relation to traffic accidents or violations, the Motor Tax Office may be asked to certify ownership of a vehicle or vehicles involved. There is a fee of 6 Euro charged to Solicitors and Insurance Companies.
Not clear as to why you you require it

Buying vehicle privately but am told registered owner is now in Oz. Have name but want to make sure. Rang Shannon but wouldn't tell me.

If you have satisfied yourself about the price and condition of the car you can ask the seller to produce a form RF101 and this will have the particulars of the vehicle and owner. This will show who the registered owner is. You will need to satisfy yourself as to the identity of the seller and that he has the authority to sell it. If you want to check the history of the vehicle and if there is any finance outstanding on it you can check on line with "Cartell" for about €25 to €35. If it is an expensive vehicle take precautions before handing over your money. Do some homework on checking out the seller.