Hey Will Monte yes, it will have it's own page so, to set it up from scratch, you would do this.
1. Go into Studio > Pages click Add New Page
2. Click the Settings button and choose the layout you want - 1 column, 2 column, etc.
3. Click Add Block and scroll to messenger and add the messenger block where you want it.
4. Click View Page and get the URL to this new page cause you will need it later. (i.e. page/NewPageName)
5. There is a trick to making the chat box bigger since I could never get it to work using custom CSS. You will need to click Add Block again. This time choose RAW block and add it to your page. Click the Raw block to edit it and make just these changes:
-- Block Layout = Content
--Inside the content box, add this code:
NOTE: You can change the height:600px; to whatever you desire but 600 is pretty good. Don't worry. this box will not display on the page 😁
Now that you have your Page built, you need to add it to your Site Navigation so members can get to it. Go into Studio > Navigation > Site Navigation
Click Add New Item. Give the new menu a System and Title name (what you want the menu item to be named). In the URL portion, paste the URL you got from View Page earlier. You should only need page/NewPageName here. Give it a icon and BAM. There you have it.