I had the same problem - only noticed when both stopped working when cold. The way to test is to get a heat gun or hair dryer and warm the faulty unit for a while (2 or 3 minutes?) and then try to start off only the faulty unit. If it starts then the problem is that the older Ducati unit develop heat related solder cracks, when the unit heats up it the joint expands and contacts and it works OK.
This was described to me by the repairers that I send mine to (In the Netherlands, I can't remember the name but google the part number and repair). It cost about $800 for the pair and took about 2 weeks - they haven't missed a beat since. Bear in mind that this is NOT a certified repair.
Beats $1600 each new.
Note that they do not dismantle the units, they drill at the right spot and the right depth to expose the offending joints, resolder then fill.
Cheers
Murray