We are UK and US citizens who are in a same-sex marriage. We are currently located in the UK. The US citizen is currently in the UK on a student visa only and not a spousal visa. Is it possible for a UK citizen and US citizen, who are of the same sex and married, to live and work in either Spain or the Netherlands where the marriage is recognised?
Yes, at least as long as the UK remains in the European Union. EU citizens enjoy a right of free movement in the EU, and family members derive a similar right when they accompany or join the EU citizen. This is governed by directive 2004/38/EC.
There are some conditions attached, the most important of which will be fulfilled as long as the EU spouse is employed.
It is not yet certain what will happen after the UK leaves the EU; this has been the subject of much debate and continues to be a topic of negotiation.
Several EU countries allow same sex marriage, and most but not all recognise same sex marriage that happened in another country. According to Wikipedia, there is a debate in the EU that all EU countries should legally recognize each other's same-sex marriages.
You should be able to move to any country in the EU where same sex marriage is legally recognised without problems. Obviously the UK leaving the EU may cause you problems (same as for all couples).