What exactly is the law regarding taking images from sites such as Splashnews and posting one or 2 onto a blog entry of your own?
The law prohibits unauthorized use in most countries. Get permission and you're safe. Wikipedia has some good articles on fair use.
My client has an issue with them contacting her over it. The thing is, that there are countless other sites using the same images without permission. My advice to her was to change the pics, she refuses based on other sites using the same ones without any apparent problem.
She doesn't really have any way to know whether the other sites have permission or not. The people who hold the copyright probably wouldn't send a DMCA take down notice unless someone informs them. The best thing to do is get permission. At the minimum read up on "fair use" to see if your client's use might fall into that category.
What I did for my websites is asking for permission for every single photo/picture I used of other sites...they were most willing to let me use them (as it is in their favour in respect to my site), but I think that most others are also willing to let you use the pictures, if only you state you got them from another website and give them credit for it.