The 1070 is doing 339 H/s and the 1060 261 H/s.
maybe you have set low intensity?
I am using Awesome Miner. I used the Diagnostics feature to view the command line and intensity is set to 64. This is the command line (user name omitted for privacy reasons):
miner.exe -di 45 --intensity 64 64 --server us-east.equihash-hub.miningpoolhub.com --port 20570 --user xxxx --pass x --api 0.0.0.0:4028
I did reach 620 H/s for the 1070 and 1060 but that's still low I think and most of the time it's between 584 and 603. I hope I can improve this as mining Zcash is pretty profitable at the moment. I'm going to tweak Afterburner settings next but my current settings work very well for Ethereum mining so I would expect them to work well for Zcash as well.
EDIT: I was underclocking the core clocks by 100 MHz or so and by changing that to zero it got me up to 656 H/s and by overclocking I'm getting even more. I have to see what that does to my power load though. Quite the difference from AMD cards where reducing the core clock doesn't affect hash rate.
EDIT2: I have hit 438 H/s on the 1070 by overclocking core by 114 Mhz. That's better! But I have to go look at the power meter to see if it's worth it.