Looking to put gutters here. 
This is how others next door are done, as well as how the ones I'm replacing were done previously. Basically a downspout connected between two gutter's via a hole cut in them.
Is this actually the correct way to do this, or is there a better way that doesn't seem so janky and leak prone.
I see they're trying to follow the line of the eaves, but the more accepted way of doing this would be a bottom outlet on the upper gutter, an elbow to attach the downspout section, then empty the downspout into the top of the lower gutter with another elbow. I'm not sure how the arrangement you show would avoid clogging and leaking. Probably not well.