How can an amateur astronomer verify the position of near Earth objects? Sometimes, hobby-astronomers use rather professional means to observe the big voids of space. Every now and then (think in months, not days) even I can locate an NEO (near Earth object).

Now, I'm assuming governments and institutions worldwide track such NEOs, have the best options to observe them, and most probably accumulated more data than we could wish for.

For hobby-astronomers who spot an NEO, it could be of interest to verify the location of the spotted NEO, and maybe even learn more about a specific NEO from additional data like name, type, path, direction, speed, first observed, etc.

Is there any publicly available, NEO-related database out there? Or is there a specific institution a hobby-astronomer can/should turn to to be able to learn more about individual NEOs?
Is there any publicly available, NEO-related database out there? Or is
  there a specific institution a hobby-astronomer can/should turn to to
  be able to learn more about individual NEOs?


Yes, there is NASA's Near Earth Object Program that catalogues all  detected NEO's and had advanced reporting and seearch capabilities (a bit overwhelming number of options actually).

There is also ESA's own (actually DLR's, the German space agency) database under the EARN (European Asteroid Research Node) project within the SSA-Programme (Space Situational Awareness) called The Near-Earth Asteroids Data Base:


  An on-line data-base of Physical and Dynamical properties NEOs, with
  the corresponding bibliographical references is available at the EARN
  site. This data-base, compiled and maintained at DLR by Gerhard Hahn,
  contains information on all known near-Earth objects
  (Atens-Apollo-Amor Asteroids), and is updated on a regular basis. It
  is also linked to, and reachable from the NEO Dynamics Site NEODyS at
  the University of Pisa, which provides extensive information on all
  dynamical aspects of NEOs, including close approach predictions and
  collision probabilities.


Honestly though, I find the NEO Program one from NASA a lot more convenient to use, even if a bit overwhelming. For extensive research though, I'd suggest referring to both of these databases, with the latter probably more conveniently used through the already linked to NEODyS Project.