Shouldn’t see any reason for any private purchasing pitfalls if you use a good independent lawyer to check out the paperwork and the property is of recent construction.
If the property is much older and requires work then please tread very carefully…which I’m sure you’re doing if you’re on this forum!
Common sense always prevails and bear in mind that vendors in Spain lie just as often as estate agents! For instance if they’ve told you that there’s no problem building another level on the house, or putting an extension on etc etc then double check every point.
You dont say where you are looking to buy and what. One thing you should look our f=o if purchasing inthe Valencia Region (and now I believe in Andalucia) is Land grab. It was meant to have changede but it is the same law in a different cover. Whatever you do whether buyingprivately or through an agent do not buy proeprty that is Suelo Rustico and please ensure you get a lawyer that is not in anyway associated with the client – best to go out of the area.
If the seller is Spanish then check to make sure the person selling it is the rightful owner – you may find there is a chain of owners because it has been handed down through the ages and no-one has bothered to register the changes. Therefore there coudl be comlications.
Also if you are buying an old property if possible get a survey done. Dont know what area yu are looking in but in the Valencia region I know of a couple of UK ARICS qualified surveyors (who are now members of the Valencia College of architects) who can do this for you. I am sure there are others. You may have a dream of doing up a place but it can easily turn into a nightmare because the work is ten times what you thugh.
Other than that it should be fairly plain sailing.
There are no more pitfalls than using an agent, infact some agents lie more than owners,(they are doing it every day 😉 ) I have bought and sold several times in spain and about 10 years ago if a seller put a property on the market through an agent the agent used to ask to see the escritura in order to forsee any problems. This has not been the case for the past five years or so. Clearly then the agents do not exactly know what they are selling so the difference between a private sale and an agent is nil. Infact you may get a better deal, I sold a house privately and gave a substantial reduction because I didn’t have to pay the 5% (plus IVA) to an agent. Do get a lawyer though.