If you can bully someone into changing their public vote and thereby changing the private votes of their multitude of followers, who cares if their single private vote was unchanged?
If someone is a politician, and a member of a party X, it seems likely that they would vote for the party X. This seems to be analogical to the public vote. Why exactly aren't we afraid that the politicians would be bullied to join the opposing party?
Probably because most people aren't politicians, and if you think that after entering politics you could be bullied into joining the party of your enemy, then you have the option of not becoming a politician. And if you don't even have that option, then the country is probably not a democracy under a meaningful definition.
To preserve this, I'd suggest the public vote to be an exception, not the rule. (Also, it should be okay to vote publicly on one topic, and abstain from voting about other topics.) By default, you didn't make a public vote. That would probably be enough so that 90% of people will never make one. So you can pretend to be one of them.
But yes, there is a legitimate concern that the easier it is to express one's political opinion, the easier it is for their employer or their family to pressure them into expressing someone else's opinion. The barrier against voluntary participation in politics also acts a barrier against such involuntary participation.
Fuck, sometimes I think that it would be best to completely fragment our identities, and pose as completely different people in different context. So there could be a professional!Viliam who doesn't have opinions about anything else than design patterns in Java, the friendly!Viliam who only posts pictures of cute kittens on social networks, and the political!Viliam who hates religion and irrationality, and no one could connect these three identities together. It would be good to have it done so easily that creating a new identity for a new context would be trivially easy.
Politics as gymnastics for rationalists. No one one Less Wrong is quite sure why politics is a taboo topic or how things got to be that way. What we do think we know is that politics is a great way to bring out the irrationality in people. So why not take advantage of that and use politics as a way to measure rationality? Since politics brings out the most irrationality, it should provide the strongest signal. Since there aren't useful objective metrics of how a political discussion went, we'd have to use subjective judgements by neutral third-party raters, kind of like they do in gymnastics. (In the comment thread for this post, feel free to find fights that you have no dog in, improvise a rationality rubric, and grade participants according to your rubric... let's see how it goes.)
Be a sheep. This is probably the exact opposite of what you were taught in your high school civics class. But if my friend Jane is more intelligent, more informed, and less ideological than I am, it seems like voting however Jane is going to vote is a strict improvement over voting however I would naively. It also saves me time, and gives Jane an incentive to put even more time in to carefully considering political issues since she now controls two votes instead of one. Done on a large scale, this could provide an interesting twist on representative democracy. Imagine a directed graph where each node represents a person and an edge is directed from person A to person B if person A is auto-copying person B's votes. There's a government computer system where you can change the person you're auto-copying votes from at any time or override an auto-copied vote with your own personal guess about what's best for society. Other than that, it's direct democracy... all bills are put before all citizens to vote on. Problems this might solve: