Should a thank you note be sent when you've received a gift as a thank you

I would say no.

If the gift is given in person, a verbal thank you (of course!) is in order. If the gift is delivered, a phone call to let the sender know it was received and appreciated should be sufficient. Otherwise, where do you stop? A thank you gift for a thank you gift for a thank you gift for a thank you gift for a thank... :)

No, a formal card is not necessary. You certainly can call and express your thoughts.