Hi Paulrep
Since you specifically asked about the "legalities", let's start with that first.
The duty under the Management of Health and Safety at Work Regulations 1999 (or other Regulations that ALSO require risk assessments) is clearly on the employer (or self-employed, when appropriate) to do a "suitable and sufficient" assessment of the risks, and, if employing five or more to record the "significant findings" identified from the risk assessment in terms of mitigating those risks to an acceptable level, which usually means "so far as reasonably practicable".
If a client brings in a contractor, this usually involves risk both to the client's own employees (and others) and also to those working for the contractor.
Hence, each (and possibly others) will have a duty to do a risk assessment to comply with the Management Regs etc.
However, what may be "suitable and sufficient" for the Client and "suitable and sufficient" for the Contractor might not be identical and in each case dependent on effective exchange of information between the duty holders - as example, the Client needs to tell the Contractor about risks that the Client puts on the Contractor, and vice-versa.
However, the Client is probably bringing in a Contractor as they are not expert in what the Contractor does and cannot be expected to be fully competent to assess the risks that the Contractor introduces to the Client's site.
Each party also has a duty under the Management Regs to have a "competent person" to assist them in enabling compliance, and if the Client doesn't know all there is to know about what the Contractor does, then I doubt that they would be competent to do the Contractor's risk assessment on the Contractor's behalf.
So, from a simple legalistic position I don't think it is a bright idea for the Client to try and take on this responsibility.
Now for the pragmatics.....
If the Contractor is reliant on some Safety bod to do all their H&S paperwork this should be sending alarm bells to the Client.
The Safety bod is unlikely to know all there is to know about what the Contractor does, partly as they are quite rarely at the scene of the prospective crime.
Which then leads to the question as to why the Contractor's line managers do not OWN health and safety AND the risk assessments, calling on the Safety bod to help but NOT to DO.
Like most other things in life Contractors tend to fall into three broad categories - the Good, the Bad and the Ugly.
The Good Contractors don't rely on Safety bods to do all their H&S paperwork tasks. The Bad and Ugly Contractors often have line managers who are often extremely willing to exhibit slopey shoulders and assume that the Safety bod will DO Safety for them, often largely ignoring the Health bits.