The subject "each of us" is singular, so if you use singular his the first time, you need to keep on using it:
It would be ruinous if one of us dropped his phone in the river, so each of us put his phone in his bag and set it aside.
And even if you use singular they, it should be:
... so each of us put their phone in their bag and set it aside.
... so each of us put their phone in their bag and we set them aside.
(So the original quote is wrong ... it appears that the author got singular and plural they confused.)