I don't think that's the solution as (correct me if I'm wrong) acoustic panels are designed to deflect/absorb sound from within the room, rather than transfer through the wall - hence recording studios use it to dampen down the sound within the room so as not to create sound-waves bouncing back into mics etc. We have had very similar problems with neighbours and the only solution we've found is air.
Stud wall with air gap between adjoining walls will be the most effective way IMO. We've tried various acoustic plasterboards (gyproc etc), fillers, silicones, sealants and sound absorbing plasterboard clips and nothing was as effective as a good old fashioned stud wall.
We had this issue with our bedroom (adjoining their bathroom) and lounge (adjoined their front door and staircase). The bedroom we are yet to tackle but it wasn't as much of a problem as the lounge (where we spend more time). We could hear conversations, arguments, doors slamming, kids running up and down the stairs - it was doing my head in.
Stud wall with acoustic plasterboard was our final option (lost 6" along one wall but it was WELL worth it). We used acoustic plasterboard because we already had it as well as wanting to be soundproof ourselves when using home theatre, plus good for insulation.