This sentence is correct. Nothing is missing.

I agree that it is unusual, but it's just about ok. I'd describe this construction as 'non-standard' ! There isn't anything missing, but if I were you, I wouldn't try to analyse it too much or try to duplicate this sentence with other words. A more common expression is to 'see+someone' + an adverbial phrase. For example 'see someone out' is to accompany your guest to the door when they are leaving, 'see someone off' is to maybe wave to them as they go on holiday, so 'see someone to their door' would be a normal expression. I guess that the 'watch' expression is an extension of this.