That's one of those things that will vary from case to case. Some folks will operate best by spending a bunch of time focusing on simply one relationship. Others won't have any trouble with multiple relationships starting up around the same time.
The thing is, that's an individual decision, for the most part. If one person requires the space to focus on just one relationship for a while when it begins, that person can do so and simply not date anybody else. That, in no fashion, obligates the other person to also abstain from dating other people; it certainly can be a topic of discussion and negotiation, though.
When speaking of various forms of non-monogamy...it ain't poly if you're just fucking around.
While polyamory, open relationships, and swinging are all distinctly different approaches to non-monogamy, they are not mutually exlusive. Folks can, and some do, engage in more than one of them at a time--and it's all good.