This needs to be split into two parts:
Only ArcGIS can edit map composition (.mxd) -- symbols, colours, labels, legend, embedded tables, etc., etc. -- and geoprocessing tools (.tbx).
However the data can be edited by any program that understands the file formats inside the package, which can be any Esri supported filetype, most common are shapefile, file-gdb, and rasters like geotiff. As the creator of the map package you have control over the file formats included. You could for instance ensure everything is a shapefile which is editable by anything calling itself GIS.
At the receiving end, unpack the .mpk file using any standard de-archiver, might have to add a .zip extension for it to be recognized, and explore the contents with the tools of choice. After the edits are done, re-zip, rename to .mpk and send back to source for review.
If the only changes are new features added/deleted from existing feature classes or raster values, the source office will see everything as soon as they open the map. New feature classes (or shapefiles or image files) will have be manually added to the map composition, but that's a straightforward process.
Best Answer
If the model is a multipatch, you could try converting it to Collada using Multipatch To Collada (Conversion). You could then view it in a number of different free software such as Google Earth.