It seems consistent, our responsibility, and authority, begins when "we" take the field, either at the recommended standard of 30 minutes before scheduled game time or some adjusted time designated by an appropriate authority.
Whenever we take the field, we have specific duties to attend to, one of which is taking common sense precautions to prevent scenarios like the sample situation from ever happening. When both teams are warming up at the same time on opposite ends of the field, establishing a reasonable buffer (to simply avoid opponents running into each other, or other common sense safety issues) seems like a logical and prudent measure.
When established, such a buffer would again, logically, be inconspicuously monitored and enforced to avoid any potential confrontations.
For those who require specific rule support to justify common sense, NFHS 1-1-6 is intended to be extremely flexible.