There are a few different aspects to the discussion about EU citizens rights after Brexit.
The legal situation quoted above (Permanent Residence Status, Citizenship etc) has not yet been agreed with the EU and is part of the Leaving Agreement. The details have largely been agreed at negotiation level, but 'nothing is agreed until everything is agreed' means that EU citizens cannot be 100% sure of their situation, particularly in the event of a No Deal, Hard Brexit. While several posters refuse to believe that there will be mass deportations of EU citizens, the position remains that if there is no Agreement, then technically EU citizens lose all of their legal rights on 29th March.
Which neatly bring up the second major aspect. It is illegal (with the threat of huge financial penalty and imprisonment) for an employer to employ someone who does not have the right to work in the UK, or for a landlord to rent a property to someone who does not have the right to live in the UK. With no Agreement, on 29th March, all of the employers and landlords with EU nationals as employees or tenants could risk being charged with acting illegally. It will be for the employer or landlord o make the necessary checks - in effect, these people become an extension of the UK Border Force.
It would be understandable if these employers and landlords decided that the risk is too great and begin to give notices of termination to their employees and tenants. While EU citizens might not be directly deported, the result would be that they would be deported by default - with no automatic rights, or employers/landlords who are fearful to engage with EU citizens in case they get it wrong, thousands would be forced to leave the UK as they lose their jobs and homes. (This also applies to their loss of rights to healthcare and education and property ownership).
A third aspect of all this affects UK citizens. If EU and non-EU citizens will be required in future to provide proof of eligibility to live, work, rent etc in the UK, then UK citizens will also be required to provide this information. ID cards, or something similar, will be required. It will not be sufficient for someone to say 'I was born and educated in the UK, so give me the job/house/treatment etc.' They will be required to provide documentary proof. [The person who almost single-handedly shot down the Labour government's proposals for a system of ID cards was one David Davis. The name may be familiar].