The true answer to this is really that nobody knows. No one has lived in a Spain where Spain has been a member of the EU and the UK has not. It will really depend on the diktats of Brussels which services are or are not provided. Dual agreements on taxation, health and other things exist between Spain and countries not in the EU as they do between UK and other non-EU countries. If UK pulls out of the EU, there will be a period of two years in which agreements can be thrashed out before UK actually leaves. It won't be a case of voting for out on Thursday and leaving the EU on Friday.
The current arrangement on pensions, for example. UK gives annual increases to pensioners living in the EU and other countries that have an agreement with UK on paying increases to their pensioners. USA, for example, does increase their pensions annually to someone living in UK so we do the same to British pensioners living in the USA. Australia does not pay these increases to Australians living in UK so we don't increase pensions for British living in Australia (otherwise we'd end up paying pensions to British living in Australia and to the Australians living in Britain). Now, if Spain (or any other EU country) does not give increases to their pensioners living in UK then it would seem unlikely UK would give British pensioners in Spain the annual increase. This is just one thing that would need to be sorted in the two years grace period.
Dual taxation would probably continue as it does with many other countries. If not, you could perhaps petition the UK government to stop paying government pensions where tax is automatically stopped in UK but that's another matter.
Not sure on healthcare as UK currently pays Spain to treat its citizens under a EU agreement so this would have to be sorted out with Brussels first (I would suggest) and then Spain which will remain part of the EU.
I am sure the current system of signing on the register of EU citizens would stop (I don't know how this would affect those currently on it) and full resident status would have to be applied for as it is for non-EU citizens at the moment. This may be automatic for those currently registered but, again, would have to be sorted out in the two years.
Those who say this or that will definitely happen or not happen seem to have access to higher levels of policy making politiical bodies than the rest of us have. It's all down to guesswork at the moment and nobody will really know the answer until it happens.