Bonkers, you can absolutely grow mushrooms in non sterile conditions - many people do - but you have to be prepared to accept the possibility of failure. It's a little like the difference between outdoor gardens and hydroponics.
I've heard of people growing shiitake outdoors in logs (with plug spawn, i.e. dowels colonized with mycelium - you drill a hole in the log, insert dowel, seal with wax, and wait - generally at least a year), never in sawdust. Oysters have been grown outdoors in sawdust, however.
For non sterile growing it's best to use low-nutrient substrates. Nutrients will increase the likelihood of competitors taking over, and I don't know of any tests that have shown them to be useful in growing wood-rotting mushrooms such as oysters and shiitake. See, however, my note about morels at the bottom of this comment.
Purchase sawdust spawn (pre-inoculated sawdust) online. You'll save yourself a massive headache over trying to start from spores, and, frankly, the cost is not that much compared to the cost of the cheapest method of starting from spores that is known to produce success in "captivity."
To get the most for your money, start by taking some of your sawdust, boiling it, and placing it in a bag with your purchased spawn; you'll get a crop of mushrooms out of that, and when the fruiting stops, you've got a big bag of sawdust spawn to inoculate your paths with. This gives you the chance to seed your paths repeatedly without buying more spawn, in case it doesn't "take" the first time.
Plus, since these conditions are pretty much ideal, if the bag doesn't fruit you'll know not to waste any more money on that strain - it's not suited to your climate or chosen substrate.
If you grow both oysters and shiitake, start the shiitake well away from the oysters - oysters seem to be much hardier and colonize things much faster, and you don't want spores from your oysters to take over everything while your shiitake are still trying to get established.
I suspect that once the shiitake are established the oysters won't be as likely to outcompete them, but I don't know for sure.
Morels are very tricky. They have successfully been grown in "captivity" but a number of factors affect their growth, including temperature and the nitrogen content of the soil. Morel cultivation is still a somewhat experimental field. Some have reported success growing morels in the garden in locations with cold winters; a few people have gotten them to fruit indoors with nitrogen supplements and an overwintering step in the freezer.
As much as I love mushrooms, my freezer space is limited enough as it is, thank you very much!