I don’t have data migration experience with OpenStreetMap, but I would like to know if there is any way to copy that public data to OpenStreetMap. Is there someone with ArcGIS experience around to see if that can be downloaded? or it is difficult to do without internal access to the system.
It isn’t quite clear what your goal is. Do you want to replace the use of ArcGis with OSM (and tools), or do you simply want to have the data in OSM as well?
There are quite few further points that will need to be consider, most not technical. But the background needs to be clear.
Importing data is subject to the import guidelines Import/Guidelines - OpenStreetMap Wiki That will sound quite difficult right now, but I would strongly suggest getting in contact with any other mappers in Ecuador, and potentially other parts of South America to discuss what makes more sense and to get help.
The technical bits and pieces of getting data out of ArcGis in to a reasonably consumable format are not that difficult, when you are ready for it.