What do you mean by "outside the watchful eye of any government authority"?
Do you just mean the ballots are filled out at home where a government authority is not looking over my shoulder? Because everything else is controlled by the government. The ballots and booklets are printed by the government (who authorize what can be on the ballot and in the booklet), are mailed by a government agency, are checked by a government authority, etc.
If anyone has the ability to confirm your vote, either without you or through you, you can be compelled or paid for it.
Imagine constructing a system that can thwart a abusive, tyrannical father who insists that his wife and children vote for a particular candidate (to make it concrete.) If you can get past him, your voting system passes the first test. Now imagine someone is offering $50 if you vote in a particular way. If there's no way to figure out how someone would claim it, it passes the second test.
The abusive father can literally just fill out all of his family's ballots, and the $50 could be claimed by filling out the ballot in front of the buyer. You could thwart this with allowing multiple votes but only accepting the first, but then the father or buyer could just have the ballots filled out immediately at the first legal moment.
I don't know that it's a thing that can be done without totally private environment around the voter and the record; meaning that the actions of the voter cannot be observed.
Those are cause for concern, but let's be realistic, the percentage of how many coercive ballots must be very low, I'd guess less than 1%. I think the pros of mail in voting (getting a greater percentage of the population to vote because they can do so at their leisure, don't have to take time off of work, don't have to stand in lines, etc.) outweigh the cons (such as potential coercion or selling of votes).
There are countries, or regions, or municipalities, or neighborhoods where the number of coerced ballots can easily be 50% or more. Voting by email is a complete no-no in those situations.
There are also countries where turnout is consistently above 70% and there is no mail voting. In the US the obstacles to voting are not having to go to a polling station: voter registration due to not having a federal ID, voting on a Tuesday rather than during the weekend, gerrymandering due to political bodies bring able to affect the redistricting process, and so on.
I was guessing that the OP meant literally looking over your shoulder. If you fill out the ballot at home, it is feasible you could be coerced to vote a particular way.
I don't think this is currently happening, so I don't think it is a major issue.
Do you just mean the ballots are filled out at home where a government authority is not looking over my shoulder? Because everything else is controlled by the government. The ballots and booklets are printed by the government (who authorize what can be on the ballot and in the booklet), are mailed by a government agency, are checked by a government authority, etc.