The vast majority of 2MASS scans will have position errors of 1
sigma = 0.5
or better using only the PPM catalog. Most scans will
have significantly better results, because the offsets become much
more accurate as the source densities go up away from the galactic
pole. Even the 0.1% of scans (46 total scans) with no PPM stars can
be tied down almost this well by using their overlap with neighboring
scans (the two east-west neighbors and the two north-south neighbors).
The 2.4% of scans with 1 or 2 PPM stars will probably also benefit from being tied to their neighbors, since the 1 or 2 PPM stars might be found on the tail of the error distribution for the PPM catalog.
However, there is a concern that using fewer than 3-4 PPM stars will make the reconstruction susceptible to one or two bad detections, confused sources, misidentifications, etc. Thus it is important to at least check how well these scans match up to their neighbors. The global refinement approach will be most useful to improve these scans.
Also, as suggested by C. Beichman, the PPM stars are often found in 2mass in only the read-1 data, which may have worse positions due to image motion problems in 50 ms integrations.
The Tycho catalog will thus certainly be sufficient to reconstruct the 2mass scans to a high degree of accuracy. Dave Monet's catalog will certainly have enough source density to reconstruct the 2MASS scans as well.
It will certainly help to use M. Weinberg's approach to do a global
solution after all the scans are obtained to rigidify the 2MASS
position reconstruction even better. However, note that the use
of the neighbors adds only about 20% more data (10% from each east-west
neighbors, and on the order of 5/350 
1% from each north-south
neighbor), so will not significantly improve most of the scans.
As mentioned above, this approach will most improve those scans
with a small number of PPM stars.