There will still be people working in each others countries. That will change little. Football of course will not be affected as it supplies a public demand.
If the UK leaves the EU, then it is far from clear what will happen to the right of free movement. The EU mandates that this right be maintained by non-members that maintain a free trade relationship with the EU - it's a package deal.
If the UK wants to make ANY change to migration from the EU post Brexit (and that desire seems to be the driving force behind the 'out' campaign), then there could be very significant impacts on UK citizens currently working in Europe - at the very least, those people will need some kind of additional work permits, and there is no guarantee that they would be granted.
As for football, prior to the Maastricht treaty, the Football League had strict limits on the number of foreign players permitted at English clubs; post Maastricht, 'foreign' was re-defined to mean 'non-EU'. If Brexit leads to this rule reverting to its previous meaning (and there is little reason to assume that it would not), then most Premier League teams will need to replace a significant fraction of their squads with UK citizens. The disruption would be enormous.