Oh my gosh at the mis-information out there on Strangles.
1. Yes, your horse can be a silent carrier of the bacteria which causes Strangles. Horses can carry the bacteria for years and shed it intermittently without showing any symptoms. This is the reason that it was long thought that the bacteria lived for years in the environment---not true, it actually doesn't survive well in the environment. At the most 2 months and that is under optimum conditions in a lab. Survival in the environment under normal conditions is only a matter of a a few weeks at best.
The way to tell if your horse is contageous is to have swabs taken from up in the nose and tested to see if the bacteria is present. That's not a 100% guarantee that your horse isn't a carrier, but it will show if your horse is shedding the bacteria currently. To determine if your horse is a silent carrier you need to have samples taken from the guttural pouch to culture to see if the bacteria is there.
2. Goats can't clear an environment of the bacteria. Sorry, it's not like the go through like little magnets or vaccuums and suck up the bacteria.
3. Vaccination does not produce long-term immunity. Vaccination generally provides limited immunity for 6-12 months. And it generally only causes less severe symptoms of the disease rather than stopping it. Long-term strong immunity occurs in 75% of horses who have recovered from Strangles.
If your horse hasn't shown any symptoms of Strangles and it's been a couple of months then it's highly unlikely that your horse is shedding the bacteria. It is still possible that your horse could be a carrier, but it's also possible that your horse has been exposed previously and has developed a good immunity to the bacteria and thus didn't show any symptoms when exposed this time because his body could handle the bacteria.