There is actually a lot of behavioral data on this question. This research is of great interest to me but I'll try not to ramble for too long. The answer is definitively no, humans are generally very poor at simulating truly random sequences. We are also poor at identifying truly random sequences. The most common error that people make when attempting to generate a random sequence is that they underestimate the number of "runs" that would be expected by chance, that is, the likelihood that the same number will be repeated multiple times in a row. In human-generated sequences, runs tend to be both too infrequent and too short. Notice, for example, that out of the three human-generated sequences which have been presented in this thread so far, two of them contain not a single run of even two numbers in a row. This is actually incredibly unlikely, statistically speaking, but that's what looks "random" to a typical human. The third contains a few runs, even a run of 3 numbers, but as others have pointed out, there are some biases in the overall frequency of particular numbers. Usually these biases are less prominent than the bias against long runs.

Psychologists in the 1970s gave these statistical intuitions about the stability of samples the tongue-in-cheek name "belief in the law of small numbers": the erroneous belief that the law of large numbers also applies to small numbers. The law of large numbers holds that as the size of a sample increases (in this case, the total sequence of numbers grows larger), the results (the frequency of each number 0-9) approach their expected values (1/10 for each number). But people have the intuition that this law, which concerns the global sequence, ought to also apply locally to any subset of the global sequence: so for example, if we have a sequence of 100 numbers, people expect that the frequency of numbers in any given 10-number subset drawn from that sequence should also be pretty close to their expected values. But that's wrong; the law of large numbers only applies to large numbers! It applies to the global sequence but it is decreasingly true for smaller subsets of the sequence. This type of reasoning causes us to err when we generate random sequences because after we list a particular number, we get the feeling that we shouldn't list it again right after, because that would make the local sequence conform less well to the expected values--in other words, it would violate the "law of small numbers."