I have a simple list with three different levels of bullets. Each level has its own style.
I want the text to remain the same size (12 pt) at each level, but the size of the bullets to decrease.
First level = 12 pt bullet
Second level = 11 pt bullet
Third level = 10 pt bullet
Whenever I try to change the size of one of the bullets, they all change. Any help would be appreciated!
More details on what you do that would be appreciated. Here’s how I would tackle the problem.
First a reminder on lists and their styling, because
Each level has its own style.
seems to indicate you do not master lists (however, their working is not obvious and there are many subtleties with them).
A list item is a paragraph who has received a paragraph style associated with a so-called list style. The paragraph style is responsible for the formatting and lay-out of text content while the list style deals with the bullet(s) or the numbering. Conceptually, there is no difference between a bulleted list and a numbered list.
Once a paragraph style can control a list (after associating it with a list style), this same paragraph style can be used for all levels of the list. The magic to change level is just to type Tab or Shift+Tab before the first character of the paragraph.
Therefore, you do not need one style per level (unless you want to change font or size but this does not result in a fine looking document). Moreover, this unicity emphasizes that the whole list is considered as a single semantic object within the document.
Now, here’s my suggestion. I’ll use paragraph style List 1 associated with list style List 1 (they are two different styles but they can bear the same name because they are stored in different namespaces; the similarity of these built-in names is not fortuitous, it is intended to remind the relationship).
To facilitate navigation, open the style side panel with F11.
Right-click on paragraph style List 1 in the style panel and choose Modify. Go to the Outline & Numbering tab. Select List 1 in the Numbering style drop-down menu. Click OK.
Click on the fifth icon from left (List Styles) on the toolbar in the styles panel.
Right-click on paragraph style List 1 in the style panel and choose Modify. Go to the Customize tab (Options in older LO versions).
A more elegant presentation than yours is to change the bullet character per level. Click on the desired level in the list Level at left, then on the Character button ... to open a character selection dialog. The default bullet is U+2022. You can use U+2219 RING_OPERATOR (white circle) for level 2, … You have nice “bullets” in the geometric shapes in the Unicode range U+25A0 to U+25FF.
To change visual aspects of the bullet, play with its Character style, by default Bullet, by default Numbering Symbols. I suggest to change for Bullets so that your bulleted lists are separated from numbered lists.
What you ask for requires you to define one character style for each level. But I would rather recommend to use a different bullet per level than changing size because this forced size will be frozen and won’t follow the size from the paragraph style should you change it.
The Position tab allows you to customize the indentation level to your liking.
Click OK.
Tab or Shift+Tab. The bullet and indent will automatically change.If this answer helped you, please accept it by clicking the check mark 
to the left and, karma permitting, upvote it. If this resolves your problem, close the question, that will help other people with the same question.
This does not answer the question. How do you change a character style? There is no “Options” tab in the “Paragraph Style: List 1” window. Is there a way to do this without applying styles? The problem is that when I modify any of the bullet sizes, all of them change.
In Pages, I can change the size of a bullet under the Bullets & Lists tab. All I am trying to do is replicate this functionality in LibreOffice.
Read carefully what I wrote: Options is in List Style: List 1.
What you try to achieve is far beyond the possibilities of “direct formatting”; you can’t avoid styles otherwise all bullets will change when you modify (globally) any attribute.
Styling is not that difficult. RTFM, this is always beneficial even if you don’t become a style professional.
Tried your suggestion and it does not work. (FYI there is no Options tab, its called Customise on my system). I guess I’ll just stick with Pages.
You’re right: the computer I used to answer had a previous LO version. On mine (with 5.3.7.2), it is also Customise. Sorry for not checking. I update my answer.