The Public Law Project has a new briefing on EU citizens’ rights during the transition period. It says:
The main takeaway is that throughout the transition period, until 31 December 2020, almost all EU rules will continue to apply in the UK. The jurisdiction of the Court of Justice of the European Union will continue until the end of the transition period. The four freedom: the freedoms of movement, services, capital and goods will continue until the end of the transition period. This means the UK will remain part of the customs union and [single] market until 31 December 2020 and British citizens will continue to be able to move freely around the EU and vice versa.
More importantly EU nationals and their family members in the UK throughout the transition period should not be asked for proof of settled or pre-settled status to access healthcare, to rent property or to gain employment until 1 January 2021. An EU, EEA or Swiss passport or national identity card, or a residence card issued by the Home Office if someone is the family member of an EU, EEA or Swiss citizen, is sufficient to show that someone is lawfully in the UK and lawfully entitled to work and rent property until the end of the transition period.