The key to determining your relationship is your Total cM which is 1019.3 cM. If you are same generation level, then according to Blaine Bettinger's Shared cM Project, the only possible relation the two of you might have is first cousin.
The Project's Table 1 Cluster Chart says that first cousins fall in Cluster #3 whose 99th percentile range is 486 cM to 1761 cM with an expected value of 850. The value you have of 1019.3 cM does not fall in any cluster other than Cluster #3. The Gen level you give of 1.9 also corresponds to first cousins.
Assuming that you are first cousins at the same generational level, that means that one of your two sets of grandparents are one of the two sets of your cousin's grandparents.
X-DNA will not help you figure that out. X-DNA is passed down from every parent to every child except that it is not passed down from father to son. Since you and your cousin are both female, you could have got your X-DNA from any of your grandparents except from each of your father's father. But he could still have been one of the grandparents in common because the X-DNA you share could have come from his spouse, your father's mother.
Assuming you are first cousins, it should be relatively simple to find out which grandparents are the grandparents in common. Find a relative you know is related to you on only one of your grandparent's sides (could be an uncle/aunt, cousin, etc.). Get them DNA tested. If they match both of you, then theirs is the grandparent in common. If not, then it's the other one.
If you are not at the same generational level, then you could be related as someone in Cluster #3. One generation apart and you might be half aunt and half niece to each other, i.e. one of you is the daughter of the other's half-sister. Two generations might be great-aunt and great-niece to each other. Via testing relatives, those can be determined as well.
Should you both have no one else to test, you can do double matching of the segments you both share with each of your matches and find triangulations that may provide you information as to how you're related. You'll have to contact some of the closer people you both triangulate with and determine how they may be connected to you.
But the X-DNA is not really a very useful part of all this in your case.