Question: What does Bernardo mean by calling Horatio and Marcellus
"the rivals of my watch"?
Answer: The word rivals used here in its primary signification of
partners, those who lived by the same rivus or stream, and
who both had use of it for purposes of irrigation.
From contentions which perhaps arose between the partners, the word came to mean those who vie with each other
Mertins, Emma. Shakespeare Examinations. Ed. William Taylor Thom, M. A. Boston: Ginn and Co., 1888. Shakespeare Online. 10 Aug. 2010. (date when you accessed the information) < http://www.shakespeare-online.com/plays/hamlet/examq/twentyone.html >.