You can do the second without doing the first; indeed, the first can be actively counterproductive to the second, since he cannot be arrested and prosecuted without first being brought within the practical reach of the US government, which the public threats to him (and the public threats to other sovereign states) have made less, not more, likely.
> Look at it from a game theory perspective; the US government 'lost' the first game to Snowden. The only reason they are still engaging with him is that the US is playing a repeating game. They need to make the penalty for leaking high enough to persuade other potential leakers to stay quiet.
Making public threats that make acheiving the stated goals of those threats less likely, thus demonstrating impotence, is not a particularly effective method of doing that.
Plus assuming that anyone would actually think that there is no desire to prosecute Snowden is foolish, as would be your suggestion for the government to keep its intentions a secret, especially given the nature of this case.
No, I'm not.
> You seem to think that keeping the desire to arrest and prosecute Snowden a secret would somehow help the government's position.
No, I said that not making the kind of public threats they are making would improve their ability to serve a particular, clearly specified national security interest, if it was their goal.
> What kind of precedent would it set if this information was kept secret and folks could then seek asylum citing a presumed yet not stated threat from their government.
People can (and do) seek asylum on those grounds now.
> When asked about it, how should the government respond?
They could point out that they don't share information about ongoing investigations.
Or that confirming or denying anything related to the matter could potentially adversely impact national security.
Or...
If you're the US government, your default assumption must be that Putin has a hard drive with everything that Snowden took. If that's the case, it makes absolute sense to publicly threaten to arrest him and charge him with the most severe of consequences.
Look at it from a game theory perspective; the US government 'lost' the first game to Snowden. The only reason they are still engaging with him is that the US is playing a repeating game. They need to make the penalty for leaking high enough to persuade other potential leakers to stay quiet.