In the period of about four months last year I visited Transnistria (a breakaway from Moldova), Abkhazia (separated from Georgia), Nagorno-Karavakh (comprising portions of Azerbaijan and historical Armenia), Kosovo (on former Serbian territory) Kurdistan (you know, in Iraq), and Northern Cyprus (Cyprus, lol) Some of these, like Transnistria and Abkhazia, enjoy considerable Russian influence. Others enjoy backing from the United States, the UN, or a nation ethnically and historically related to theirs.
Kosovo currently has 114 diplomatic recognitions as an independent state. Serbia, from which Kosovo was more or less carved, does not recognize it as a sovereign state, but has begun to normalize diplomatic relations with its government per the Brussels Agreement of 2013.
Most of the people I spoke to in Kosovo identify themselves as being Albanian (the nation state of Albania is adjacent to Kosovo's western border) and indeed, 88% of the country is ethnically Albanian. The border passing from Albania to Kosovo was pretty much just a checkpoint with some amiable guards.
Transnistria, or officially the Priendestrovian Moldavian Republic, is recognized only by the partially-recognized states of Nagorno-Karabakh, Abkhazia and South Ossetia. It's the most Soviet place I've ever been, with various Soviet symbols and decorations present throughout. Although it's not officially recognized by Russia, there are most certainly Russian troops and tanks posted on the road from Tiraspol to the Moldovan capital of Chișinău, and are apparently in Transistria as "peacekeepers."
Hey, I'm sorry. This is a really interesting question and one I've been struggling with for months now. But I have significantly underestimated the huge political and historical complexity of many of these places seeking to be recognized as independent nations and I won't be able to do the proper research for a long while, because I have managed my time poorly as of late.
Despite having visited these places, I'm only equipped with a rudimentary opinion of how to address this question. If you re-asked it during Alumni week, you'd definitely get opinions. But don't do that, necessarily, unless you want like, a paskillion answers. Maybe.
But here's a couple of my thoughts (prepare to insert foot in mouth):
I would have liked Kurdistan to have become an independent. Why does the Sykes-Picot agreement of WWI, over 100 years ago, dictate the freedoms of current groups? That's not really that helpful.
I think places like Abkhazia and Transnistria should be recognized as independent entities. Do I condone the bloody wars and, in some cases genocide, that have aided these coming into being? No, but they are currently autonomous, whether or not the Euro-centric European Union for political purposes has decided to recognized them as such. Also, the UN is sometimes viewed as being, like, impartial, but they are punks too in many cases.
It's interesting that Kosovo didn't have much backing from the UN until the USA decided to official-ize them.
Taiwan is totally a country. They don't consider themselves part of the People's Republic of China, regardless of whether or not they are claimed by their powerful neighbor.
For Nagorno-Karabakh, now Republic of Artsakh, I feel they have some legitimate claims to land they occupy. Azerbaijan also has legitimate claims to a lot of that land. Both sides have done some pretty bad stuff trying to get it or keep it, like when Armenian (ethnically) people completely destroyed and depopulated Agdam, a city once inhabited by 39,000 people, mostly Azeri. It's probably the largest ghost city in the world, twice the size of Chernobyl.
Anyways. I'm just rambling, not really answering. It's really fun to read Wikipedia's list of states with limited recognition and then start delving into the history of each. There's some interesting stuff happening in this world.
Finally, I don't think All People will agree on the political legitimacy of one nation over another. Case in point: the Holy Land. Also, any Facebook or Youtube comment zone about any of the places I have mentioned. There's a lot of strong feelings and the answer is rarely simple.
For some discussions of this that address it better than I, you might enjoy:
P.S. Wikipedia's list of ghost towns and cities of the world is another good time.