First of all I can not repeat it too many times, don't be in a hurry to start running after pregnancy. Give your body time to heal and recover. Make sure that you postpartum stomach is strong enough and fully recovered. Running is very challenging for the core and pelvic floor. You don't want to make diastasis recti worse or get some kind of pelvic organ prolapse, that can happen if you start too early!
But if you are fully recovered and strong enough to start running, hopefully after months and months of power walking (I think you should wait at least six months after the delivery and perhaps even longer if you are breastfeeding) you might want to do it while taking care of your baby and run with the stroller. I have done that quite a lot and I experienced that holding on to the stroller handlebar actually provided me with some extra stability since your upper body and arms are more fixed than if you are running without the stroller.
So is it necessary to buy a special jogging stroller for the purpose? Well, I can definitely see the benefits with a stroller designed to use when running, especially if you are a runner and will be doing it a lot. But perhaps you are like me, someone who runs now and then to get some variation in exercise. If that's the case you probably don't want to spend all that extra money to get a special jogging stroller, or you might not have the space to keep more than one stroller, if that's the case you will probably be relieved to hear that it isn't absolutely necessary. I kept running with my regular stroller - a bugaboo camelon - and it worked perfectly fine. Clearly I didn't run on rough terrain. I mostly stayed on city streets, but it worked fine in other places as well, as long as the ground was rather flat. Also, before you go out and spend a lot of money on a new stroller, go for a test run, perhaps running with a stroller isn't your thing at all.
Good luck!
//Nicole Hellgren
This is the stroller I use when I run with my baby and it works fine. However, I do prefer to run without the stroller...