If you weer to add a spoonful of yogurt to cottage cheese would the bacteria multiply/colonize the tub in a few days?
The trouble is, in order for anything close to that to happen, you'd have to leave the container at room temperature. The fridge keeps bacteria from multiplying.
However, as you know, leaving it a room temp would also allow the really nasty, bad-for-you bacteria to grow alongside the good ones.
Now is that because both are ready made products making them more prone to spoilage or is it just a possible concern?
I make my own Kefir and have left the grains in a quart of milk for up to 3 days removed the grains and then left the quart out for another 2 days. Doing this I have never had an issue.
As TC mentioned, refrigerating the concoction isn't optimal for culturing as Lactobacillus species (the classic "good bacteria" of fermented foods) prefer substantially higher temperatures (~37C).
Beyond that, and in theory, the issue with living a little and letting your concoction sit unrefrigerated isn't so much with regards to food poisoning (the characteristic acidification resulting from microbial metabolism is known to deter the growth of pathogenic bacteria, and while that nasty E. coli O157:H7 has been shown to be acid-tolerant, initial pasteurization of both products weeds it out efficiently), but in that cottage cheese is lacking in the growth substrate lactose.
Why? Because the manufacturing of cottage cheese starts with the incorporation of Lactobacillus species (e.g., L. lactis) into milk; here, a decreased pH brought about by lactose fermentation results in the denaturation and precipitation of casein proteins to form the curds we end up eating (a process augmented an enzyme complex called rennet). While whey and the sugars remain soluble, further expelling of these nutrients is both desired and accomplished by washing the curds with heated water; these washings result in bacterial strain loss/attenuation (which is why cottage cheese isn't classically considered to be a great probiotic food).
Now, I wrote "in theory" above for two reasons: 1) pasteurization is NOT an infallible process and I don't want you to try this, shit your spleen out and narc on me to your doctor/lawyer, and 2) my understanding is that many large-scale commercial cottage cheeses add cream or milk to the curds in the final stages of the manufacturing process (for whatever reason), so lactose may be present in various brands.
So, if you feel so inclined, as an alternative you might consider shopping for brands of cottage cheese that advertise "bioactive/live cultures" (or whatever) as these products are the result of either gentler methods of manufacturing or inoculated post-production (the preferred route as this allows for a more accurate/precise control of bacterial content).
The problem is the mystery bacteria that can float in and take up residence in your milk or cottage cheese.
Spoilage is a separate, naturally occurring process that results from the lactobacilli metabolizing the lactose in the milk. It's not inherently harmful to eat such products, but like I said, the danger comes from the uninvited microbes that can float in.
I posted my answer before I saw yours! Yours is better!