I can talk about Ireland, as I've been an election observer there. The way we do it is that ballot boxes are locked and sealed with tamper-evident seals after a polling station has closed. Elections observers; including representatives from the political parties may request to add their own seals. In some particularly contentious districts this is done, but for the most part people are happy with the official seals.
The ballot boxes are then transported by the police force to the nearest "count center". The next morning, the seals are inspected and those ballot boxes are opened. All of the ballot boxes in a constituency are counted together in a secure, but open area. Here's a flickr set with a good number of photos showing how it's done:
https://www.flickr.com/photos/redmum/sets/72157600270850764
the counters are within the fenced area, and the observers - including many people from the political parties, surround the fence. The entire process is easy to see.
One particularly important part is what happens when the boxes are opened. The contents are just dumped out on the table and one by one each vote is turned to face up and towards the observers. The observers then "tally" the votes and mark which candidate (or referendum choice) the voter marked as their first preference.
All parties participate in this tally and it provides the first take on what the result will be. The margin of error on the tally is Contentious votes with identifying or ambiguous marks and so on are kept aside and argued over by people like me for an hour or so, but they never make much of a difference.
The end result is a process is very verifiable and auditable, in easy-to-understand human ways; you can literally show up at a count center and count the votes yourself as they come out of the boxes, and make sure that you're not being duped. That's a nice accessibility property too.