Sick days aren't a real solution because people only take the days that they are the most sick off. But will happily work while contagious and 'just starting' to feel sick.
In my personal experience companies have always said something like that, in particular in email or other forms of recordable communications, but then don't really back it up. Employees come in obviously sick their boss says "are you sure you should be here today?" but subtly indicates their approval for being in the office.
Companies need to move to actively disciplining employees who come in sick instead of either working from home or taking paid sick days.