If the opposing Pokemon were to use Echoed Voice in the same turn my Pokemon uses Echoed Voice, would the slower Pokemon's attack get the base power raised? Or would it have no effect since the Pokemon isn't on the same team?
No it does not gain a boost. I tested this on Pokemon showdown and it doesn't gain a boost if the opponent uses echoed-voice.
NOTE: I would show the replay but they aren't working.
Hope this helps!