I need to create a sharing button such as the one of Twitter. Are there any standard sizes I should use for it?
(Edit: I'm developing a sharing service)
There exist no standard for this. Only common sizes as DA01 mention in his comment.
The reason you find sizes such as 16, 32, 64 etc. comes from the old days when most programmers also did the icon designs. The size 2^n fits nice and tight with how a computer operates memory wise (memory alignment) and programmers choose these sizes for that reason, as back then this gave a small optimization in performance but also in file size as there where no need to pad image lines of these sizes with data to fit the alignments used (memory is aligned on 4 bytes segments, 8 on 64-bit systems).
Today performance and byte size is not an issue with small images like these.
So my suggestion is that you use a size that esthetically fits your design. Don't look at the pixel numbers but see how it actually look and feel in your design.
A little more detail would help. There are two possible answers to your question.
You're developing a sharing service and you want your icon to work well next to other service icons.
In this case you'll want to provide standard square icon sizes: 16px, 24, 32, 48, 72, 96 ... I think that's about as big as a share button will ever get. A social service like ShareThis would be a good reference on the standard implementations.
You're designing custom icons for your site's share links. In this case, you either work with a stock set that fits your brand or you design your own at any size you please.