Main reason to split is so if some thing causes the supply to trip, you don't loose all sockets, with a typical 60 amp supply to a home, unlikely you would draw more current than one circuit could supply, but could have more leakage to earth than a single 30 mA RCD can reliably hold in at, the limit is considered as 9 mA. It would need to be a rather large home for this to be a problem, and step one would be all RCBO (MCB and RCD combined) before looking at splitting, and since should one ring final fail you would not want extension leads up/down stairs, better to be split side to side than up down.
If the total load is a problem then better to remove some heavy loads,
Appendix 15 BS 7671:2008 said:
The load current in any part of the circuit should be unlikely to exceed for long periods the current-carrying capacity of the cable (Regulation 433.1.5 refers). This can generally be achieved by:
(i) locating socket-outlets to provide reasonable sharing of the load around the ring
(ii) not supplying immersion heaters, comprehensive electric space heating or loads of a similar profile from the ring circuit
(iii) connecting cookers, ovens and hobs with a rated power exceeding 2 kW on their own dedicated radial circuit
(iv) taking account of the total floor area being served. (Historically, limit of 100 m² has been adopted.)
It was the 100 m² bit which resulted in two ring finals, the volt drop permitted was I think 4% which from memory was considered to means 88 meters of cable limit on a ring final, this changed to 5% power and 3% lighting so ring final limit increased to 106 meters of cable.
Near impossible to measure the length of cable in meters, but one can measure the loop impedance or prospective short circuit current since voltage fixed at 230 volt, in real terms both show the same. So with an incoming supply with 0.35 Ω mid way around the ring final we should measure less than 0.94 Ω if the volt drop is within the 11.5 volt permitted with a 32 amp ring final. This assumes a design current Ib of 26 amp, that is 20 amp at centre and 12 amp even drawn throughout the ring final so not written in stone.
Whole idea is to keep the amount of cable used to a minimum. The less cable the better the loop impedance, and the 1/3 rule on drilling beams resulted in more cable being used, having one ring final with all socket points going down and another with them all going up means the one where the points go down uses more cable so a better loop impedance up stairs to down stairs, so again better split side to side so both rings have an even load and loop impedance.
But as an after thought not easy, so likely best option is to run a kitchen or utility room ring to remove heavy loads like dish washer, washing machine, tumble drier, oven, kettle etc from the general ring final.