Generally, throughout the books and the Show, I don't remember people from Essos (or some other place besides Westeros) feeling anxious about long winters. We know that they have tropic climates and dress way more lightly than people in Westeros; this doesn't mean that they don't have long winters, but so far we haven't seen people worry about them. But if there were long winters there too, wouldn't dry and hot places (like Slaver's Bay) have turned to places with temperate weather?
Essos is pretty big, so it might be that the northern areas (like the Dothraki Sea) might have longer winters than the southern ones. We don't know that much about Essos' history to make a clear statement. Most of the POVs we've seen are on the south side of Essos, so it's natural not to have heard much of winter in those areas.
As far as the Others go, when they first came, during the Long Night, they came from the land of always Winter, which is in Westeros, right North of the Wall. So far they have caused trouble to the Westerosi. We don't know yet if they have some kind of base somewhere else on the map and plan to (or already have) attacked other places.
If I remember correctly, GRRM once said that Westeros is the only continent that stretches that far north, so we couldn't assume that Essos is connected with the North some way off map. So, the Others would have trouble getting to Essos; we don't know if they can swim or sail (as of yet).
If we go back in time, the Long Night is something that happened to the whole Known World (including Essos), so we can see that Essos isn't excluded from every 'magic' event that takes place in the world. During that time, the east side of Essos, especially the areas of Rhoyne had their rivers frozen.
TL;DR: We don't know much about the others yet; we don't know what woke many centuries ago and we don't know why they were sleeping for so many years, until now. So, we cannot know for sure how they can (or will) affect Essos.