1. There are well-settled copyright principles that hold a viewing event to be a "public performance" even though a single person only is doing the viewing at any given time. A leading case involved a video-store owner who owned one copy of each film he had in stock and who set up private viewing rooms in his establishment in which he allowed a patron to view a film privately that the patron had rented from the store. This enabled the store owner to buy a copyrighted video once and to set up a business in which he repeatedly displayed that video to multiple members of the public, one at a time. In that case, the court held that the showing of the one copy of the film owned by the video store owner repeatedly to different members of the public constituted a "public performance" (the case, Columbia Pictures v. Redd Horne, may be found here: http://scholar.google.com/scholar_case?case=1713962666875062...). This case is offline law but the principle it articulates is clear. If copyright holders are in business to sell or license their copyrighted works, and if someone buys one copy of such work and sets up a business open to the public to make money from repeated showings of such work (albeit one at a time), the copyright owners are having the value of their works taken from them by the intermediary business owner, who gets to profit from such works without having obtained any right or license from the copyright owners to license such works to others. A leading copyright treatise (Nimmer) sums this up, then, by saying "if the same copy . . . of a given work is repeatedly played (i.e., 'performed') by different members of the public, albeit at different times, this constitutes a 'public' performance." (discussed here: http://laboratorium.net/archive/2011/03/16/that_zediva_thing...). In this offline context, then, it is indisputable that you can have a "public" performance for copyright-law purposes from what is technically a private viewing.
2. Logically, the one-copy rule should not apply to the case where I own my copy and merely choose to display it for my private viewing (or private listening in the case of music) through various methods of playback, including by use of a cloud service. And this is what both Google and Amazon are banking on in the way they have set up their cloud music services. Their theory is that the service they offer does nothing more than allow you to upload a song you already own and access it from different browsers and devices. To avoid the one-copy rule, neither Google nor Amazon "de-duplicate" user files, which means that users will literally access the exact files that they themselves uploaded into the service (meaning, therefore, that millions of copies of the same file may exist in the same cloud).
3. The big test case that is pending in the courts is EMI v. MP3Tunes, and this case will test whether the old offline rules should apply literally to the digital world. That is, MP3Tunes is a music locker that does de-duplicate its files, i.e., stores one copy only of each song (rather than one copy for each file any user has uploaded) and uses that copy to enable users of the service to replay songs they have uploaded into the service (though not literally from the exact file that the user uploaded). It is this case - where the intermediary business owner is using "one copy" only to enable multiple members of the public to listen to the same song, albeit only for their own private listening - that the EMI lawyer is commenting upon when he says that one person listening to his own music via the cloud constitutes a "public performance." Though tone-deaf on how this must sound to the average person, the lawyer is basically reciting what the offline caselaw held in connection with the one-copy rule. This may indeed be irrational as applied to the digital world (since, in fact, the service is requiring each individual who listens to upload his individually-owned version of that song, which distinguishes this case from the video store owner who bought only one copy in total), but this is why he is calling it what he does.
4. EFF has a good discussion of some of the key issues here: https://www.eff.org/deeplinks/archive (see under the heading "Do music locker services violate current copyright laws?").
The fact that important web-based services are having to use such inefficient means as storing millions of versions of the same file just to deal with current copyright laws shows that is is high-time such laws were revamped for the digital age. The last major revision was in the 1970s. The laws in this area made sense in their day (at least for those who don't oppose IP rights of this type) but are today strained to the breaking point.