I know that the Fx-6300 would overheat and cause system failsafe crashes with a stocker cooler, trying to game, stream and video chat. If you're going to be doing many intensive loads at once, your stock cooler may not be able to handle it, even with AMD's newer coolers (with the heat pipes). If you're not going to be pushing the processor to the limits for prolonged periods of time, then you're fine.
I'd really suggest a Hyper 212 Evo though. I've got one on my 8320 and it keeps her below 50c during prime 95. For $35, I'd say it's worth the peace of mind and quality cooling, even for just stock usage (no OC).