Does Engi Medi-bot dispersal heal the boarding party?
There have been a lot of times where I've thought about buying a Teleporter while using the Engi ship (or obtaining the Engi Medi-bot dispersal augment elsewhere).
Will the healing from the augment apply to my boarding party while it is in the enemy ship?
No, the Engi Medi-Bot Dispersal ONLY heals crew members on YOUR ship. This means they will not be passively healed when boarding enemy ships. Only the crew that is currently onboard your ship receive the passive health-regen.
As RavenDreamer points out, Mind-Controlled allies will not receive the passive effects from the Medi-Bot Dispersal because during Mind Control, the controlled target acts like an "enemy".