How do you know if your starter is powered mainly by yeast or bacteria?
Unfortunately many kinds of fermentation produce CO2 as a byproduct, so the presence of bubbles hardly give you more information than 'it is alive'.

If what you want is a precise identification of what strain of yeast and bacteria are present in your starter, I see no easier way than looking under a microscope or making a laboratory analysis.

If you just want to heuristically determine if yeast or bacteria is present, the easier way is to guess from the taste and smell profile of your starter. The guess can be more or less polite depending on your training, just like in wine tasting.

For example: 


alcohol smell hints for the presence of yeasts, the most common agents responsible for the alcoholic fermentation.
vinagre smell is associated to acetic acid, which is the subproduct of the fermentation of glucose by bacteria of the AAB (acetic acid) family. 
acetone (nail polisher) smell is usually associated to the presence of ethyl acetate, the esterification of ethanol + acetic acid. Thus my guess here would be the presence of both bacteria from the AAB family and yeast, which produces the ethanol via the usual alcoholic fermentation.
sour taste / smell (like yoghurt) is usually associated to lactic acid, byproduct of lactic fermentation which hints the presence of bacteria from the LAB family.


These are the most obvious examples, but you could go fancier and try also to identify more subtle notes like specific fruits tones associated to some esters, etc. A good place to learn about those things are the ressources in brewing websites, specially in what concerns wild beer (lambic) - a process somehow similar to cultivating a sourdough starter. See for example this thread or this wiki.

Note that it is very likely that what you actually have is a combination of these processes. And as the acetone smell example shows, the combination of two processes can lead to more than the simple sum of their byproducts. It is also important to keep in mind that some of these molecules can produced by both bacterias and yeasts (like ethanol, although yeasts are usually more efficient). This is what makes a sourdough culture complex and unique. Changing the hydration of your culture, you also change the balance of bacterias and yeasts, since certain strains are more or less hydrophilic.