A meeting in Parliament on Tuesday heard the EU referendum result had left many people feeling threatened about their futures.
The event was told this applied to both the estimated three million EU citizens, including some 900,000 Polish nationals, currently living and working in the EU and the one million UK citizens who live in member states. This includes about 300,000 Brits in Spain alone.
Two Spanish-based experts on constitutional and international law, who were asked by Parliament to draft a report on how citizens' right might be affected by Brexit, presented the findings to a workshop.
Diego Lopez Garrido, who specialises in constitutional law, told the meeting there was widespread uncertainty about what will happen to citizens' rights, including their property and residency rights, when the UK formally withdraws from the EU.
He said this is expected to happen two years after UK Prime Minister Theresa May triggers article 50, which is due to take place next week.
But he said EU and British negotiators need to address the rights issue as a matter of urgency, adding, "We have heard a lot about the trade and economic consequences of Brexit but, once formal negotiations actually start this really has to be the number one priority because there is great uncertainty at present for millions of people."
Garrido, who helped draft the report on rights, said, "People desperately want to know what is going to happen to their jobs, their pensions and their homes when the UK leaves the EU. They also want to know this as soon as possible.
"The fear is that Brexit will have a negative impact on the lives of EU citizens in the UK and British people in Europe which is why negotiators have to address this now."
The right to residency, he said, was the core issue concerning such citizens.
"It has been asked if this is an acquired right but the answer is no, it is not. The fact is that, once the UK leaves the EU the whole body of EU law will cease to apply in Britain and the existing rights of affected citizens will cease to exist."
The meeting was organised to debate the constitutional relationship between the EU and UK and consequences of the referendum last June.
Garrido said that any agreement thrashed out by the EU and UK had to comprise the issue of citizens' rights, "otherwise, millions will be left in a legal limbo" after Brexit happens.
This includes EU citizens "who went to live and work in the UK in good faith and in the belief that they would enjoy a stable future. All of a sudden they see that this is no longer the case. They feel threatened by Brexit."
He pointed out that the legal situation for such citizens remains the same until 29 March 2019, the date when the UK is due to formally withdraw from the EU.
But he added, "It could be that, after this date, citizens will start to take their cases concerning their rights to court."
It is possible, he argued, that citizens could use the European convention on human rights as a defence to uphold their family and residency rights.
He said, "There is case law for this and it is why the agreement that is drawn up has to be a good agreement and is as close as possible to the current situation. This is what the EU and UK should strive to achieve."
Garrido, in outlining the effects of Brexit, also pointed to possible "imbalances" between the rights of EU citizens in the UK and British people in Europe.
He said, "At present, citizens of the 27 member states who continue to live in the UK will have fewer residency and property rights when Britain withdraws than British people who will still live and work in Europe who will enjoy a more comfortable position. This is because the charter of fundamental rights will still apply to UK citizens in the EU.
"EU citizens will find that their rights are substantially reduced post Brexit. This relates to the migration issue which underpins the British government's Brexit objectives."
He added, "The big problem, though, is that this is going to create an imbalance between the two sets of citizens."