Let's consider the classic example: Australia, which has used STV/IRV for more than a century.
For calculations of votes, we need to consider each electorate separately. Calculation of the global outcome (seats won etc.) is trivial.
It should be noted that in any voting system, tallying the final result given "the number of votes each party received" is a technical process and is rarely controversial. The process is more complicated for STV, but it's still pure math. Controversy may arise at the earlier stages: how exactly ballots are counted, what is considered informal (invalid) vote, how ballots are presented, etc. This is where most scrutiny is needed.
Generally, for the public, the results are presented already calculated and ranked. They still present the "number of votes" for each candidate, but this number is calculated in a less trivial manner than just adding up votes. Still, the process of calculation is also presented for those who are interested.
Here is an example from the most recent federal elections for the electorate of Adelaide. (This is the lower house election, which is IRV - a single winner version of STV. It is a bit easier to demonstrate, but the principle is the same. Note that Senate is still elected by candidates, not parties; just there are multiple winners in the last round and the threshold is lower than 50%).
The (abbreviated) "final" table looks rather "normal":
The catch is that there are only two candidates shown out of 7! This is because other candidates were eliminated in the STV counting process, and their votes were redistributed to these candidates. This is essentially the result of the last round of counting. In Australia, this is often called Two party preferred or Two candidate preferred result.
Next thing presented (and often publicised) is the First preference count for each candidate. This is the exact number of ballots which gave first preference ("1") to this candidate, like if it were "traditional" elections. We can see that for these two candidates, the count is different and much lower:
In STV, this first preference count has largely psychological/symbolic value. In this example, we can see that a third of votes for the ultimate winner Steve came from other parties, whereas Amy had much higher share of "personal" votes. Of course, it is a well-known feature of STV that the final winner may not be the one who had the most first-preference votes.
Nevertheless, there is an "implementation detail" in Australia that parties/candidates who get enough first-preference votes (a fairly small percentage) get certain monetary compensation for the election expenses. This is to avoid a flood of spurious candidates who might register just "for fun".
The next thing one may wish to examine is the Full distribution of preferences table on the same page. There, each round of counting is presented, starting from first preference and culminating in the two candidate preferred result. It is clearly observable how, for example, the 870 votes from the first-eliminated candidate are distributed to the remaining 6 candidates in the second round, and so on.
Here, by the way, is where Senate election with its multi-winner count gets more complicated. There, if a candidate gets more votes than the quota requires (at any round), the "excess" votes are taken away and redistributed to other candidates with the proportional weight. As a result, each elected candidate gets exactly the same number of votes in the final round, and everyone else gets 0! This is not required for IRV as such process will not change the ultimate winner. The calculation is also published, but it is a multi-page PDF document (example of a smaller one for Tasmania - 30 pages).
For those who want to scrutinise further, AEC publishes the results in machine-readable formats as well.