As the negotiations progress, we still unfortunately lack clarity from the UK government. However, as people cannot simply wait and see we are forced to speculate based on available information and statements by those involved.
One important thing to remember is that freedom of movement is not derived from the EU, it's actually part of the European Economic Area (EEA) rules. So while the UK voted to leave the EU, it did not vote on membership of the EEA and due to issues like the Irish border and Gibraltar it may be difficult to leave it, so full freedom of movement may continue.
Are UK citizens working in the EU affected?
The EU seems committed to looking after EU citizens, including British ones even after Brexit. The EU has designated individual rights as a high priority and even if the UK walked away with no deal, it seems likely from statements by key people such as Guy Verhofstadt that UK citizen's already in the EU exercising their EEA treaty rights would be protected. Individual nations, such as Spain, have also said they will not require anyone to leave, without setting out detailed proposals.
In the worst case such people would be forced to return to the UK, if they could not secure residency rights in an EU country. In that case there are other issues that have not even been considered, since both sides want to avoid this situation, such as if their families would be allowed to go with them (assuming they wanted to live in the UK).
Another very bad but entirely possible outcome would be for UK citizens to become "land-locked", that is able to remain in the EU country they are in but unable to move to another one as EU citizens can. This could be extremely problematic for them, because many EU companies do cross-border business regularly.
One other potential consequence is a flood of elderly and/or unemployed expats back to the UK post Brexit, putting a great strain on local authorities and the NHS. For example, if they had retired to Spain but then lost their right to remain there.
The Irish and Gibraltar/Spain borders are another big unknown. Many people cross them daily for work. Some UK politicians are calling for an end to this free movement, which would presumably make them unemployed at the very least. For Gibraltar the very worse case would be having to rely on food aid deliveries from the UK by air.
Are EU citizens working in the UK affected?
The UK government has said that they can stay, but it isn't clear what rights they will retain yet. For example, if they need to visit family for an extended period in another country, will their residency rights in the UK lapse?
Should the UK walk away from negotiations with the EU, the government has said it would not deport EU citizens already here, but there is no legal mechanism to prevent them reneging.
Are non-EU citizens working in the UK affected?
The UK government has mentioned introducing a "points based system" for immigration post-Brexit. It has also promised to reduce net immigration to the "tens of thousands", which would severely impact non-EU citizens trying to reunite their families or people trying to hire foreign staff.
To give you an idea, there are already tens of thousands of EU family reunion immigrants every year, so if the government stuck to its promise to EU citizens there would be have to be a complete end to all other forms of immigration. Similarly, UK universities are reliant on non-EU students for income, and their numbers would exceed this promise too.
Given this is seems likely that, while not as a direct result of Brexit since the UK already has full control of immigration for non-EU citizens, things will get a lot worse for non-EU citizens. The ones already here may find it much more difficult to bring family members, or to visit family members, or to obtain permanent residency rights in the UK.
What other consequences might 'Brexit' have on EU/UK citizens?
This is a rather broad question, but in relation to residency rights the other group who will be negatively affected are families. Currently if a British citizen works in, say, France, they can have their family join them from anywhere in the world. If they eventually return to the UK, their family can go with them, regardless of UK immigration rules. That right may be lost, making working in EU countries much less attractive since even if they have less draconian immigration systems the UK citizen might be forced to abandon their family if they themselves must return to the UK.