"Just a Hint," by David Brin. First published in Analog (April 27, 1981). I encountered it when I bought a secondhand copy of a collection of several of Brin's early shorter pieces of science fiction (as opposed to his early novels): The River of Time.
The basic premise of the plot is just as you described. The point-of-view shifts back and forth between two scientists, light-years apart, who each are worried sick about a problem that the other one's civilization has already analyzed and solved . . . but they don't know this, and it looks like they never will, due to lack of funding for radio astronomy projects such as SETI (and the alien equivalent). And on each world, there are disasters looming on the horizon which mean it may be a very, very long time before anyone revives the science of radio astronomy -- and that's only if some of their descendants live long enough for it to ever become an option in a rebuilt civilization, you understand.
Sam Federman is concerned about the possibility of nuclear war, and wonders if aliens, with a completely different perspective, might be able to see a new approach to resolving geopolitical problems -- that is to say, "new" to human psychology, but blindingly obvious from the alien viewpoint. It particularly irks him that the environment here on Earth has been largely cleaned up after we realized the cancerous side-effects of massive industrial pollution, but now we may not last long enough to enjoy it. If only he could get a hint from some intelligent, impartial outsiders who didn't share all the cultural paradigms that human kids subconsciously absorb as they are growing up on Earth . . .
Meanwhile, on another world with a similar level of technological development, Federman's counterpart, an Academician called Fetham, is looking at things from a different perspective. He, too, wishes he could get plenty of funding for efforts to make radio contact across the light-years with some alien culture that might have a very useful perspective. I'll quote a few nonconsecutive paragraphs of his musings to show beyond any reasonable doubt that this is, in fact, the same story you remembered.
After the invention of atomic weapons, before he was born, his
parents' generation had finally found the motivation to do the obvious
and abolish war. The method had been there all along, but no one had
been sufficiently motivated before. Now the fruits of peace were
multiplying throughout the world.
But the Plague had then come among them, soon after the last war, and
now affected almost everyone. Lung ailments, skin cancer . . . that
horrible sickness that struck the mercury and bismuth mines . . . the
death of the fisheries.
Huge sums were spent to find the microorganisms responsible for this
rash of diseases. Some were found, but no germs yet that could account
for the wide range of calamities. Some scientists were now suggesting
a pathogen smaller than a virus.
Fetham, too, speculates that if long-term funding for his work were assured, it might be possible to contact some other sentient race which had already encountered and somehow overcome similar problems, and who might be kind enough to share the secret of how they'd managed that remarkable feat!
The last few lines of the story (still showing us Fetham's thoughts):
Suddenly he had a totally irrelevant thought.
I wonder where the birds are? They used to be all over this part of the city. I never noticed that they had gone, until now.
"I suppose," he sighed. "I suppose I was hoping for just a hint . . ."
