In eBird, every complete checklist should be thought of as an attempt to record everything that you can detect from where you are standing or walking. If you are standing along the banks of a river you should certainly scan the river and both of its banks. If you are along a ridgetop, you should be counting birds that you can see in all directions. However, at times there may be a geopolitical border that bisects your walking path or your field of view. Birds never have cared much about geopolitics, and in many cases they freely cross borders. What is the best way to do your eBirding in cases like this?
Within eBird, a bird "counts" if if can be seen from where you are standing. This applies to yard lists and locality lists (which have almost always used these rules). It also should apply to your eBird checklists whenever possible. This means that if you are standing at Salineño, Texas, and you see a Lineated Woodpecker flying over the trees on the opposite side (thus in Mexico), this "counts" in eBird for your U.S. and Texas list. Our reviewers should also validate the record. However, in cases like this where you are seeing a very rare bird in another country, state, or county, please do make it clear in your checklist comments where you saw the bird. Lineated Woodpecker still does not have a confirmed record for the United States, so having these notes would be important for those using the data.
Duplicate entry as Incidental is OK. In eBird it is also OK to report birds seen on the opposite side of a border. If you go back and check your woodpecker photos and realize it was in fact a Pileated Woodpecker, you'd want to get that in the official record as a first country record for Mexico. In this case, it is OK to use this hotpot for La Gloria, across from Salineño, to report your bird which will then show up on your Mexico and Tamaulipas list. However, these should always be reported as Incidental checklists (and thus, not reporting all species) since you were not on foot in the area and could not do a complete survey of the birds seen at the site.
The reason for this is because eBird works best when users designate their lists as complete checklists. We do not want birds being left off your lists simply because it was across an arbitrary (to the bird) boundary.
Note: Some birders care deeply about their state and local lists. If reporting a bird on the complete checklist from where you are standing is not something you wish to do, then we recommend considering your checklist "incomplete" (not reporting all species) since you are intentionally excluding a bird that you saw on your survey. This is OK, and in this case you would report one checklist using a Traveling or Stationary protocol (with distance, duration, and start time) but indicating that not all species were reported; the bird of interest would be left off this list. For the second list, plotted on the other side of the border, you would report as Incidental.