I figured, since a lot of people are bringing up the Birthday Problem, I'd add a separate answer just for discussing it. Simply put for those not familiar with it, the idea is that based on probability the actual number of individuals needed within a sample before overlapping becomes nearly certain is far less than the actual number of possibilities.
The original problem discusses how many people you need in a group before you are almost certain to have two people with the same birthday. Obviously 366 people means that it's 100% certain (discounting leap years), but 70 people gives a 99.9% certainty of at least two people having the same birthday, and even just 23 people gives roughly a 50% chance.
So let's apply this to the current problem. The first question, though, is how we should divide the world to check it. The most logical way to do this is via population samples, since geographical area obviously won't have equal weight due to variable population density. Let's use 40 million, as that's around the size of most of the largest metropolitan areas of the world (Tokyo, NYC, etc.). Given 7.6 billion people, this would give us 190.
Using a calculator, this would give us a surprisingly high 96% probability of at least two of our 35 people sharing the same population segment. Now, this may seem impressive, but one must consider that 40 million people is the population of many Indian states, or any 2-30 American states (with a few exceptions such as New York or California). "There might be two people who are both in California" isn't too helpful, "There might be two people somewhere in the American Midwest (sans Illinois/Michigan)" is less helpful, and "There might be two people in all of Canada" is even less so.
So let's reduce our search to within a unit of 10 million people. That's about the same as Tokyo proper or all of the country of Greece. Now it's a little more specific. Reducing our population by a factor of 4 increases our number of sections by the same. So now we have 760 possibilities. This brings our odds to about 55%. 783
Still larger than might be expected, but it's now a coin flip as to whether anyone is in the same city proper, or section of a populous state.
Now, that's not to say that it's a coin flip whether someone is in the same population sample as any other given person. If you were a survivor, it would not be a coin flip that a second person in 10 million also survived in your area, but rather a coin flip that there is a second person anywhere. There's a 16% chance that a second person is in the same 40 million original population section as you (1-(189/190)^34), and 4% chance that there's another person in your own sample of 10 million.
Now, though, let's consider this by area. For example, if by "randomly scattered" the OP means "picked up and dropped in random locations". (The Birthday Problem requires equal probability, so this approach isn't really applicable to realistic distributions.) Given roads and long views in rural areas, let's consider the world to be one fifth as large as it is. This helps reduce the effect of deserts, mountains, and other boundaries, as well as the effects of people travelling on roads. (Let's call this our "scaling factor")
So we have a land area of around 150 million square kilometers, reduced to 30 million square kilometers from our scaling factor. Let's say that you can find someone within 30 kilometers, about the size of a large city, when they fire a gun, honk a car horn, etc. This gives us an area of 900 square kilometers to locate someone within. Let's call it 1000 to make things nice and even.
30 million over 1000 gives us 30,000 land units. The odds of two people both "spawning" in any single land unit of 1000 square kilometers would be about 2%. This is actually surprisingly high, but still means that it is unlikely that anyone would start within 30 kilometers of each other. Though the odds of two people "spawning" within 100 kilometers of each other (about 10 times more area) is actually only about 18%, which is amusingly high. Though again, this is anyone, period, not the odds of a you having a partner in your 100x100 km area (which would be around 1%). 783