No.
It does not matter where your website images are stored.

It would only matter if the images are slow to be downloaded, thus slowing down the site.

Not sure about the SEO, but when I was an astronaut, I discovered redundancy is a lifesaver

I am with Dario Ferreira .  I know that photos may slow down the site, however in luxury photos make the site.  It works for us and our clients.  A

A slow loading or non-mobile compliant website will pay a really big SEO penalty.

There are format and technologies solutions that do not necessarily require a hosting website. However, having backups somewhere is always a great idea.

On my IDX pages, I store them on Pinterest.

The only benefit is to page speed load if this other site is a content delivery network (CDN).  Those are often MUCH faster than what you can do and the content gets copied around the country or world (depending on your plan) so that visitors get the images from the closest server to them.

If you can get a subdomain like "CDN.YourSite.com" it's better for SEO than safhasofhsa2348329.cloudfront.net or "YoursiteCDN.com" as any links to the photos will go back to your root domain.

Same as Dario said.

I store on my own site.  I make sure to resize and compress photos so it does not bog down the site and load time. 

I'm late to the party, but was about to say the same thing as Dario.

And, I'll add that if page speed is the issue, there are ways to adapt that w/ better hosting, CDN and image compression.  I do not embed my images.  and, I do leverage the alt tags in them.

If the external server has a higher bandwidth it would be beneficial. Other than that, I don't know any other reason.

You have many great answers here.

I’m with Dario Ferreira 

Good question, it may vary by what site you have them on.

No idea.

Great question Dave Martin (703) 585-4687 I will be eager to see the experts like Debbie Gartner answer this one.

I wouldn't think so.

I keep my pics on a separate drive

Beyond my paygrade. Parked and reading.

I have professional photos of each listing on a memory stick.... I have one stick on each laptop.... 

No the images need to be labeled. home #1, home #2 so the scrubber can digest and rank the quality of images.

Nope. Other than slower than death loads when lights dim in your house or an entire community is knocked off grid into blackness. There is a benefit from adding your blog post image to pinterest with the "havest" from the post not just a pin upload. Red rover red rover bring it in and over from the blog post url to appear for SEO value. Linkage is a good thing. People search under image, video, map, etc and use the other tabs to the right of just website url option too. So you want to be showing up on top and in that first screen full ta da for all the information vending machine selection options.

I host them on my website and add the link in other places.

No idea 

Never gave this any thought and not sure where you store pictures has any bearing.