As a resident of Chile on holiday in Sweden (90 days tourist no visa required), will I get a passport stamp upon leaving Sweden even though the flight is to another Schengen country, say France or Spain?
No, you will not pass through the systematic passport check when flying between Schengen countries, and you will not get passport stamps at either end of the flights.
In some, more-or-less exceptional, situation you may encounter "extraordinary" passport checks in or between the Schengen countries. Even if you run into such a check, you are not crossing the external Schengen border, so you wont't get a stamp (falsely) claiming you have done so.
Flights between Schengen countries are regarded as domestic flights, so there will not be any stamp. In fact, you may even not be controlled at all.