No, native speakers do not intentionally drop any apostrophes. This is one of the most common mistakes that English speakers make when writing: not knowing where to put apostrophes. In English, the rule is that apostrophes are for two purposes: (1) possessives and (2) contractions. Here are some examples:
1) "It was Jason's horse that you were riding." Here, the apostrophe indicates possession/ownership. The horse belongs to Jason.
2) "When you're walking across a road, you'd need to watch for traffic." Here, the apostrophe is used to represent contractions. "You're" is contracted from "you are" and "you'd" is contracted from "you would."
However, this can create confusion when both contraction and possession can apply to the same pronunciation of two words. The best examples are "its" vs. "it's" (as you brought up) and "who's" vs. "whose." In these two cases, both apostrophe words are contractions ("it's" is short for "it is," and "who's" is short for "who is") because English also has independent possessive words that don't require apostrophes to indicate possession: my, your, his, their, etc. "Its" and "whose" are both independent possessive words.
Recently, I've noticed another mistake becoming more prominent among English speakers: they're putting apostrophes before the plural "-s." For instance, sentences like this are becoming common:
"There were several plane's parked on the tarmac at the airport."
This is an incorrect usage of the apostrophe. "Plane" in plural form is "planes." "Plane's" would mean something belonging to the plane. I don't know why this mistake has become so widespread lately, but it needs to stop.
I hope I've helped answer your question. Please let me know if anything confuses you.