Are there rules about when a player may use a chess engine to aid his game? Is it always frowned upon? Do you have to specially ask? Not that you can stop your opponent from using an engine, but is it cheating, or is it ever acceptable? Thinking about it, I'm not sure what the point of two players using engines would be.
If you haven't agreed before the game then you should assume that it's not allowed. It will usually be regarded as cheating. If you wish to use a computer you should ask your opponent before you do so.
Playing with the assistance of a computer is sometimes called Advanced Chess. If you agree before hand that it's allowed, then there's no problem and both can use a computer freely. Some Advanced Chess tournaments are also played with standard time controls.
Thinking about it, I'm not sure what the point of two players using engines would be.
The point of Advanced Chess is that a computer is strong at fast and accurate calculation but weak at strategy, while the human is stronger at strategy but more prone to errors in complex tactic situations. The human can use their strategic ability to guide the computer into strong strategical positions or won end game positions, while the computer can check that there are no blunders that the human may have missed. If played well, the playing strength of the human-computer team can be higher than each individual playing alone.
In earlier times we used to analyze actual positions of correspondence chess games of a club member to help him win his games. Nowadays it is more common to ask the computer instead to check the validy of positions. Therefore in my opinion nothing has changed, except that the computer is more of a tactical genius than the common club player. Although there is some truth in the saying 'chess is all tactics', I think most corrospondence chess players use chess engines mostly for checking against tactical loopholes, while using databases to check for common motives or win probabilites. But for the strategy part of the game, it is the human mind that wins the day (as long as chess is not fully calculatable from start to end). Except for playing strength levels where cheap tactics always decide a game, the help of a computer is not a replacement for thinking yourself. I know of several very good correspondence chess players which would gladly play any player who restricts himself to using chess engine moves, because they expect to beat such an opponent easily. The moment this won't be true anymore correspondence chess will be dead (or at the very least be reduced to a clumsily slow chess engine testing ground).
Rules will usually try to regulate the allowed use of computers. The earlier practice of adjourned games was abandoned because of computer help becoming available, not the least because tablebases allowed perfect endgame play. In tournament halls the use of a cell phone or chess computer will often lose you the game instantly. In club matches the frequent use of the toilet room with a computer will be considered cheating, too. I remember some experiments with chess players being allowed the use of a computer during the game ("advanced chess"), but it never became popular. As long as you do not register the computer as a player (instead as yourself), the use of a database, cheat slip or chess engine is probably everywhere considered cheating. In the 80s there were some tournaments where chess computers where allowed to enter, and I remember dimly that former world championship candidate Robert Huebner forfeited a game once because he refused to play against a computer in such a tournament.
Extremely critical is the use of computers during online play, because no one can see what you are doing in person (like setting up a second computer to calculate moves for you, while "you" play a game on the first computer). Most playing sites have special rooms where the help of computer is allowed - everywhere else this is considered cheating. Of course, the use of a computer is a must to play online at all, but only as a human interface device (for inputting and receiving moves). Mind you can often register a chess engine as a separate player, so everyone playing it knows he is playing against a computer. But if you state you are playing yourself although you use a chess engine I'd consider it cheating.
First, two players using engines happens to compare the strength of engines (or the computers they're running on, but usually the first option). This is typically arranged as a computer chess tournament, or a "free for all" type match, where it is explicitly allowed to use engines.
I know that a lot of the chess websites actually have detection algorithms, where they compare your moves to those of popular engines, if people flag you as a potential cheater. Usually, the idea is to not use an engine, so that you're actually competing human to human. Often, in correspondence chess, it is acceptable to refer to books (and sometimes game databases), but not analysis engines.