AS already said, && operator means, that BOTH conditions has to be met. So in your condition it means that you want worktype type to by freelanceand fulltime at the same time, which is not possible :)
Most probably you want employees that have work type freelance OR fulltime, thus your condition should be:
people.Where(w=>w.worktype=="freelance" || w.worktype =="fulltime")
Or, if person can be set more than once in this table, then you could do:
people
.Where(w=>w.worktype=="freelance" || w.worktype =="fulltime")
// here I assume that you have name of a person,
// Basically, here I group by person
.GroupBy(p => p.Name)
// Here we check if any person has two entries,
// but you have to be careful here, as if person has two entries
// with worktype freelance or two entries with fulltime, it
// will pass condition as well.
.Where(grp => grp.Count() == 2)
.Select(grp => grp.FirstOrDefault());
&&to "or"||if I'm understanding your goal? What you currently have looks like a guaranteed empty set.