The general answer is, yes, cover can affect both parties. However, there may well be some circumstances where one party can claim a bonus from the cover, but the other can't. The best example for this is a Low Obstacle, which the attacker explicitly ignores if he is closer to the cover than his target. The other way this is likely to happen is if an attacker is standing at the corner of a wall, targeting someone in the open - he can generally shoot out with no penalty, but the defender would likely have the cover penalty when firing back. This is because the person firing chooses one corner of his square as the origin of the attack, but then has to trace a line to all corners of the target's square, and if any of them cross a piece of cover, the attack considers cover.
To understand this, you probably need a diagram - if you look at the diagram on page 253 of the Starfinder Core rulebook, it covers SOME situations, but not all. The example also appears to treat the Obozaya vs ksarik situation a little differently to what the text says (i.e. lines passing ALONG a border don't provide cover, only those CROSSING a border).
If you view the diagram as if the ksarik was not present, and the characters are all targeting each other with ranged attacks, then Obozaya will have cover from Altronus, but Altronus will not have cover from Obozaya, due to Obozaya basically poking around the corner. Raia and Obozaya have clear lines of sight to each other, so no cover applies. Raia and Altronus cannot see each other, so each has total cover from the other.
You can also see a more complete diagram at d2opfsrd. However, note that Pathfinder has a couple of slightly different rules for cover compared to Starfinder. In Pathfinder, ranged attacks all work out cover using the same mechanics as Starfinder. However, melee attacks from an adjacent enemy work differently, and instead of choosing a corner for the attack to originate from, melee attacks check from every corner of the attacker to every corner of the defender.
So, in your particular example, it actually depends on how the object you're hiding behind looks and is defined. Being at chest-height, it's not likely to be considered a Low Obstacle. If the crate is only a single square in size, you will be able to fire at the goblin without cover if he is not directly in front of the crate. If the crate is wider than one square, and you are at one of the edges of the crate, you MIGHT be able to fire at the goblin without cover if he is on the same side as the edge. But in most other circumstances, the goblin will also have cover from your shot.
