First of all, that depends on the probability of winning the lottery. Many lotteries have probabilities that are so ridiculously unlikely that 500,000 plays means nothing. Seriously, it's not uncommon to have odds of 1:100,000,000 and therefore 500,000 isn't even enough to, on average, expect 1 winner, let alone 2.
However if your lottery has a win rate of 1:100 then winning twice would be akin to rolling a dice with 100*100 =10,000 face (1 for each combination). And for 500,000 tries or 250,000 two roll tries you'd expect to see that on average 25 times.
Now "on average" is a weasely concept, because these are independent events and it's just the average of large numbers of trials that are predictable, but a single event or even a low number of trials could happen more or less often than that.
The expectation value just says that with N tries and a probability of 1/N you can expect to see the result once "on average". However it doesn't say whether it's on the first or last try or twice in the first N rounds and none in the second and so on.
So for normal lotteries your "conspiracy theorist" would have a strong point that odds of 10^16 or 1:10,000,000,000,000,000 are so unlikely that a rigged process is more likely than a fair game. Though in the end unlikely doesn't mean impossible, which is why statistics has to manage two kinds of errors one where you accept the hypothesis (that the game is fair) despite it being unfair and the other where you reject the hypothesis despite it being fair.
And where you set these limits is essentially arbitrary and more of an expression of what you'd like to avoid more as shrinking the margin of error on one increases it for the other. So for example for tests for diseases you'd rather have false positives than false negatives. So better more people who are marked as diseased and can be further tested than having people marked as healthy that are actually sick and could spread the disease without knowing. While if you produce idk high quality products you might rather reject good products than risk having a faulty one on the market. But again with 10^16 against you even conservative error margins would probably reject that.
In terms of what sounds more likely to be rigged, well I guess it's more of a psychological thing. Like while correlation doesn't equal causation and while short term trends don't have to be long term trends, we still have a tendency for pattern matching and if things happen at the same time or if we see a trend we are likely to assume a connection or a larger pattern (which is what bites gamblers quite hard).
Like in the case of winning the first 2 lotteries it's a winning streak of 100% that we suppose will continue. While in the second it's winning 2 out of 502 lotteries which is a better than expected win ratio but still a pretty bad one. Like if you started playing now it would, on average, take you a year to win twice even with these odds being massively more in your favor than the actual ones.
Though if you want to rig a lottery - and let's be real all lotteries ARE rigged against you - you wouldn't do it by winning multiple times you'd just set the jackpot slightly lower than ticket price * expectation value. Because that way you actually win the lottery with every game because on average you net more than you give out. And that's something that you can plan with some reliability (for large numbers), while the individual game is pure chance.