For me to get a specific audio format, do my A/V receiver and DVD player both need the decoder?

Answer 1

No only one or the other.

If the receiver has the Dolby Digital and DTS decoders then all you need is to hook up the player with a optical or coax digital cable to the receiver

If on the other hand you have an older receiver that has no digital inputs but has a multi-channel input and a dvd-audio player with 6 analog outs for surround sound then you can use that type of connection and the player will decode the Dolby and DTS surround.

Answer 2

The AV receiver is more important.
If you want to listen to DD 5.1 or DTS, then your player just passes through that stream to the receiver and the receiver has to be able to decode it.

Your player should be able to decode 2-channels out of the Dolby Digital stream.