Vaccination reactions are allergic reactions which are highly individual inflammatory responses against specific proteins entering the body. Reactions this bad are thankfully quite rare.
Your dog cannot "catch" the reaction from the other dog in any way, so it is totally safe playing, sharing bowls, etc.
Please let your sister know she MUST tell any future veterinarian she sees about her dog's bad reaction (make sure she knows to tell them directly every time, not just rely on their records) as they may need to take pretreatment precautions for any future treatments given, depending on what they are.
The vaccination, or the vaccine reaction's are not contagious.
Your puppy may be susceptible to the same reaction when given the distemper vaccine. Find out if the vaccine given your sisters dog contained Leptospirosis, since this is the most common vaccine reaction culprit. You can request a distemper combination vaccine without Leptospirosis.