I'm solidly on the liberal side of most issues, and I definitely try to remain informed, which includes exploring conservative viewpoints and media.
The problem is that MOST media (of any focus) is shallow. Reading the "opposing" side is just as tiresome as reading points I already know if obvious information is skipped over or not explored (actually, more tiresome). I end up chasing down lots of details myself, which is time-consuming and frankly, not really my primary skillset. It theoretically is the skillset of, um, journalists.
So when on the left I see "Immigrants on average commit less crime than native-born in the US" and on the right I see "3% of the population (immigrants) commit over 50% of the crime", I'm not finding any media exploring both sides. I can spend a few hours chasing down this one fact (a pretty important one, but still just one), which leads me to find vague references to a DOJ report on reported crimes supporting the conservative view, and surveys of self-reported crimes supporting the liberal view. So neither position is without basis, but I really have no better info on which is more true, and if I spend all this time chasing down evidence myself, the media is serving zero purpose.
Secondly, I've found left-leaning sites that do more factual digging (at least it seems that way) and will cover and attempt to disprove some of the opposing arguments. (Both sides have vapid overly dramatic coverage offered by many sources - I'm referring to the better sources on both sides). The closest I've found on the right is the National Review. I'm all ears if someone has better suggestions, but as it is I tend to read more within my "echo chamber" because it's actually the best source of nuanced information I've found so far.
Thirdly, when you're reading two sides: One side says things that you mostly know and agree with. The other side says things that are often ridiculous and false on their face (as far as you are concerned). Why would you want to frustrate yourself with the state of humanity all the time? I read enough (I think) to get an idea of the zeitgeist of others, I read enough to understand what the basis of their arguments are, but any reading past that tends to be far more frustrating than enlightening. (I assume this is something true for both sides).
Lastly, left and right may both have reasonable people, but the right seems to have a lot more [carefully edits this description several times] general disdain for science and more embracing of hypocrisy. The Left has baseless GMO fears, anti-vaxers, and sometimes more optimism than might be best, but in general I can expect less denial of well-demonstrated concepts and science. (I'm sure this is slanted by bias, but I'm happy to go point by point offline if anyone wants). This means that while I "check-in" with sources espousing opinions I don't agree with, I have no more desire to spend a lot of time there than I do on a site informing me that eating GMO corn will cause me to mutate.
Off topic side note: I really hate how the useful GMO discussions (monocultures, sustainability, environmental impact, prions, etc) can't really get any progress because everyone spends all their time rebuffing ridiculous accusations. Just like "what do we do about climate change" doesn't get as much discussion as "is there actually human-caused climate change".