Spectre
Owners of AMD and ARM systems shouldn't rest easy, though, and that's thanks to Spectre. Spectre is a more general attack, based on a wider range of speculative execution features. The paper describes using speculation around, for example, array bounds checks and branches instructions to leak information, with proof-of-concept attacks being successful on AMD, ARM, and Intel systems. Spectre attacks can be used both to leak information from the kernel to user programs, but also from virtualization hypervisors to guest systems.
Moreover, Spectre doesn't offer any straightforward solution. Speculation is essential to high performance processors, and while there may be limited ways to block certain certain kinds of speculative execution, general techniques that will defend against any information leakage due to speculative execution aren't known.
Sensitive pieces of code could be amended to include "serializing instructions"—instructions that force the processor to wait for all outstanding memory reads and writes to finish (and hence prevent any speculation based on those reads and writes)—that prevent most kinds of speculation from occurring. ARM has introduced just such an instruction in response to Spectre, and x86 processors from Intel and AMD already have several. But these instructions would have to be very carefully placed, with no easy way of identifying the correct placement.
In the immediate term, it looks like most systems will shortly have patches for Meltdown. At least for Linux and Windows, these patches allow end-users to opt out if they would prefer. The most vulnerable users are probably cloud service providers; Meltdown and Spectre can both in principle be used to further attacks against hypervisors, making it easier for malicious user to break out of their virtual machines.
For typical desktop users, the risk is arguably less significant. While both Meltdown and Spectre can have value in expanding the scope of an existing flaw, neither one is sufficient on its own to, for example, break out of a Web browser.
Longer term, we'd expect a future Intel architecture to offer some kind of a fix, either by avoiding speculation around this kind of problematic memory access, or making the memory access permission checks faster so that this time interval between reading kernel memory, and checking that the process has permission to read kernel memory, is eliminated.