Well, there are many parents that do it and never cause any damage to their baby's health.
IMO Infants under 6 mo should only be in a jogging stroller in conjunction with a car seat or bassinet that is made to be used with that stroller. Some strollers come with infant inserts and recline fully for infants as well.
You should not actually run with your newborn either, even when secured in a car seat or bassinet. Their neck muscles are not strong enough to withstand the shocks properly. Most jogging strollers point this out in the fine print of their instruction manuals.
I cringe when I see tiny newborns propped up in an umbrella or jogging stroller that does not recline properly, nor allows for them to be harnessed securely. The injuries that can be done by this range from whiplash to traumatic brain injury, strangulation and falls. Now this may not happen very often, but just think of what it would do to you if it did happen to your child.
My sister used a side by side double jogger for kids two years apart - she thought it was not very maneuverable and tended to pull to one side because of the difference in weight. An in-line solution like the phil & teds doubles kit may work better for actual running.