I was under the impression that Unit testing in an agile development was the same as for a traditional one - although you will do it more often.
Different processes state the need and method for writing unit tests a bit differently. XP, for example, encourages test-driven development in which you write the unit tests before the corresponding production code. Another process could encourage writing unit tests after writing the code. Most process instances in use probably don't say explicitly when the unit tests should be written (or how often). Besides, a process can be agile even without unit testing.
Shouldn't every developer be unit testing?
Certainly.