What are the prime challenges in search for extraterrestrial intelligence? What have we been able to accomplish so far and what are the prime obstacles that block our sight? My understanding is that Radio Telescopes are the prime tool for the search. I wonder what kind of challenges the astronomers who work on it are facing.
We have no idea how much we’ve been able to accomplish so far in the search for extraterrestrial intelligence.
If there isn’t extraterrestrial intelligence then our progress is zero.  If there is extraterrestrial intelligence and we already found some, then we’ve made 100% progress!
I think the major obstacles for SETI are threefold: funding, stigma, and physics.
Funding:  This article is a couple years old, but it does a nice job of summing up historical low funding for SETI.  Most of the funding is private and on the order of millions to tens of millions of dollars a year.  That might sound like a lot, but ALMA cost $1.3b.  A few million a year isn’t enough to sustain a very robust program or buy, maintain, and staff an observatory.
Stigma: At least when I was working for the NRAO, SETI work wasn’t considered serious astronomy.  There was some smirking in the lunchroom from some of the astronomers towards the folks “looking for little green men.”  I think it’s more useful to try to improve telescopes and data processing techniques, but I don’t think anyone should be stigmatized for pursuing their intellectual curiosity. Stigma usually means less funding, less access to highly sought after observation hours on the best telescopes, and less people choosing to join the field.
Physics: Radio signals decay in power as the distance squared and even the nearest star, Proxima Centauri, is 4.3 light years or about 271,937 AU away from our solar system.  That means they would need an enormously powerful emitter and we would need a very sensitive detector to detect their signal.  See: how far away could we detect that Earth has life?.