What generation watches are you having this issue on? Are you replacing the entire assembly or just the glass/digitizer?
I've replaced a good number of screens on the first gen and Series 1 watches and so far I've found out a few things:
1. You can't replace just the digitizer unless you also have the service cable and can flash the firmware. I've heard you can replace just the digitizer and then run an update, but the timing on this makes it difficult and I've never tried it.
2. Some Apple Watches just don't like certain screens. I've seen this happen a lot when I have pulled assemblies from a parts donor and put them on the watch to be repaired. The donor assembly either won't display anything at all, will get stuck in a boot loop, get stuck in the flashing charging cable loop, or touch doesn't work. So far I've had pretty good luck with disconnecting and reconnecting the battery, swapping for a new battery (probably the same as a disconnect though), and running an update with the new screen in place. But again, the timing need for the update can be an issue. Also, there's something to be said for doing a force restart several times. I've had a couple instances where that brought functions back slowly or all at once.
I haven't found a common denominator for the above problem yet, but I haven't gone back to the donor watches to compare OS versions, etc. It doesn't seem to matter if you are crossing parts between first gen and Series 1, or if you are staying with identical models.
I've only done a few Series 2 and one Series 3 so far. Other than the adhesive being a lot stronger, they are all around easier to deal with and I haven't had any issues with them.