For each puzzle grid, the main slitherlink puzzle refers to the clues that have bigger numbers while the secondary slitherlink puzzle refers to the clues in the black circles.
Starting deductions for the main slitherlink on the left leads to the following. The secondary slitherlink is also filled in.
Then, initial deductions for the main slitherlink on the right will result in the following:
Then, focusing on the '3' in R8C6 for the main slitherlink on the left, the top segment cannot go up or to the right. If it does, then there is no way to satisfy the '3' for the secondary slitherlink in R7C7. Therefore, it must go down joining up with the segment at the bottom.
After that, the '3' in R7C7 for the main slitherlink on the right must have 2 line segments meeting at the vertex of R8C6. The bottom segment cannot go down or right as it would then share an edge with the other slitherlink. So, it must go up. The grid at this point:
Following that, the loop segment on the right side of the main slitherlink on the left is completely blocked off from turning left. So, it must extend up all the way until it reaches the third row. This, in turn, prevents the loop segment of the other slitherlink in the bottom right corner from turning right. So, it must go up and then turn left. The grid at this point:
Now, focus on the four '3's in the upper right corner. The upper right '3' of this group of four cannot go to the right as it will then share an edge with the secondary slitherlink. Therefore, it must go to the left and this allows us to resolve the loops around this region.
After that, the loop segment at the top of the main slitherlink on the right needs to escape via the left towards the '1'. It must go through the top of the '1' like so:
Then, we turn our attention to the main slitherlink on the left. The '1' clue in R1C3 cannot be satisfied by a loop segment going through the top or right side. If the loop segment is on the left, then the '2' will have 3 loop segments surrounding it. Therefore, the loop segment must be on the bottom side. This segment must then connect to the loop segment coming in from the right side. At the same time, the loop segment must exit from the left side and satisfy the constraints of the '2' in R1C2. Once that is done, the loop must then snake towards the '3' in R3C1. This leads to the following grid:
Again, we focus on the main slitherlink on the left. The bottom edge of the '2' in R4C4 cannot be a loop segment as it will cause the '1' below to have 2 loop segments. So, the top and right of that '2' clue must be filled. Following some knock-on deductions, we get:
Still, on the same slitherlink, observe the loop segment coming near the '3' in R8C6. That loop segment runs into a dead-end if it goes left, so it must go up. After passing the right side of the '1' in R7C4, it cannot turn right as it would then be forced to share an edge with the secondary slitherlink. So, it is forced to go up all the way to meet the loop segment surrounding the '2' in R4C4.
For the final steps, we now look at the main slitherlink on the right. The loop segment near the '1' in R7C5 cannot go down, and so it must extend to the left, like so:
After that, look at the '1' in R8C2. If the top side of that clue was to be filled, then it will share an edge with the other slitherlink. Therefore, the loop segment is on the right side of that '1' and we can complete the main slitherlink on the right by connecting this segment to the main loop segment.
Then, it is an easy task to solve the other slitherlink. After cleaning up, the final solution looks like: