Looking at the image below, let's pretend that we are seeing a series of blog posts on a website. The first post was written in Arabic and the second in English. The user viewing the website primarily speaks Arabic but also speaks English. The rest of the website is displayed in a right-to-left layout orientation in Arabic. Is it correct to display the RTL language as RTL and the LTR language as LTR? Or should they both be consistent?
Yes, it's correct to display based on the language direction, and if you're going both ways, then keep the general design center-focused and let the languages go in the direction they go in. Ultimately right-to-left languages like arabic draw the eye on the right versus left-to-right languages. If content types switch, no problem; readers will adjust based on what they're reading.
Remember that we've all learned to read (and if someone is complaining about the direction...they're either intolerant towards other languages or don't actually know how to read), and that direction is really step 1 to any content. That's why all major english-speaking websites start at the top left. Just make sure that your website mimics the dual-directional languages that content is written in.
As an Arabic speaker, when I am on Arab websites that contain mixed language content, if I see Arabic in LTR or English in RTL, I feel that not enough effort has been put into doing the site.
Your display of Arabic in RTL and English in LTR showing each language in its correct format shows that effort and thought has been put into this and would increase my trust in your site.
One option is to offer a language preference so the entire screen/site is a consistent. Or try a language translator (google?).
If those are not feasible, the direction should reflect how its read by the native speaker. Try not to trade usability (legibility) for the sake of aesthetics.
Leverage headings and other typographical elements to help lead the reader's eye down the page. Also try to balance the visual density of the paragraph blocks with consistent letter/character heights, try to make them equal in weight. Skim this article for further info. Read this article about Writing For the Web for more info.
If by chance you are using Wordpress, try this link ideas.