I have written a previous post on how to get Blogger to host your photo for the profile, sidebar, <img> tags, etc at Getting Blogger to host your pictures for the profile, sidebar, etc.. However, tests have shown this method no longer works.
For New Blogger, this is what you have to do. Upload your photo to a post which if you don't want it to be on the main page, pre-date it to 2005 or something, or even in a blog created specially to host pictures (Blogger does not limit the number of blogs you can create). I upload this photo as an example:

Click on the picture (or the caption) to get an enlarged version of the photo. In the method describe previously, you will use the URL in the address bar of this page. Don't. It wouldn't work now. You can try it yourself if you want. Try putting that URL into the Photo URL in the EDIT PROFILE page and confirm for yourself that it can't be done. You will get the error message "Cannot find file at specified URL: link is broken (failed request)" and if you have already put in your profile photo, that will remain as this attempt will fail to replace it. You can also try putting that photo into the sidebar using "Add a Page Element" in the Layout, choosing "Picture" in the pop-up, and putting that URL in "Configure Image" page. You will get the error message "Please specify an image". Putting that URL int the <img src="photo URL"> tag will also not display a photo.
Instead, in the page containing the enlarged photo, in the menu bar, click VIEW > PAGE SOURCE to get the source code for the page. In this case, the source code is as given in the scroll box below:
<html>
<head>
<title>copyright+notice.jpg (image)</title>
<script type="text/javascript">
<!--
if (top.location != self.location) top.location = self.location;
// -->
</script>
</head>
<body bgcolor="#ffffff" text="#000000">
<img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhulzuD6Xa-ljbAjrrBZfXKValCO-18o1FXQdPo08JxuNyQ_GSvDnOfYozVU_B9apNMe40WmXF3w2x9VFBEvMtxsn3CakzfMxlMw-B6vRfusIyRNIRF9pyw62SJveLoy65i7zkumf8flAE/s1600/copyright+notice.jpg" alt="[copyright+notice.jpg]" border=0>
</body>
</html>
<head>
<title>copyright+notice.jpg (image)</title>
<script type="text/javascript">
<!--
if (top.location != self.location) top.location = self.location;
// -->
</script>
</head>
<body bgcolor="#ffffff" text="#000000">
<img src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhulzuD6Xa-ljbAjrrBZfXKValCO-18o1FXQdPo08JxuNyQ_GSvDnOfYozVU_B9apNMe40WmXF3w2x9VFBEvMtxsn3CakzfMxlMw-B6vRfusIyRNIRF9pyw62SJveLoy65i7zkumf8flAE/s1600/copyright+notice.jpg" alt="[copyright+notice.jpg]" border=0>
</body>
</html>
And almost at the end of the codes, you will see src="http://1h6.google.com/.....
I have to put the full URL in a scroll box, otherwise the long unbroken URL will extend into the right sidebar (FireFox) or cause the sidebar to slide to the bottom of the page:
https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhulzuD6Xa-ljbAjrrBZfXKValCO-18o1FXQdPo08JxuNyQ_GSvDnOfYozVU_B9apNMe40WmXF3w2x9VFBEvMtxsn3CakzfMxlMw-B6vRfusIyRNIRF9pyw62SJveLoy65i7zkumf8flAE/s1600/copyright+notice.jpg
This should be the photo URL you should use for the profile photo, for inclusion in the <img src="photo URL"> tag, for Picture in the sidebar, etc.
Update 12 April 2007: I have made the actual photo URL in the first scrollbox red so it is easier to spot.
Incidentally, the above photo is copyright of Tourism Malaysia in which I have put a copyright notice at the bottom right of the photo. I will describe how to put a copyright notice in your photo in a post in Dummies Guide to Google Blogger. (I have just added that post to that blog. You can read it here: A simple way to put a copyright notice to your photos
The photo can be used to promote tourism to Malaysia. 2007 is Visit Malaysia Year, and many programs have been planned for visitors and this will be a good year to visit Malaysia. For more information, see Guide to Malaysia.
Acknowledgement: This tip was provided by my online friend sookietex of Public Domain Clipart