Community Hospitals usually do both step up and step down care. The medical care is usually given by a particular Practice or by the hospital's own dedicated team of GPs employed by the CCG.
I have worked in 2 and we took calls from GPs who wanted to avoid acute hospital admission but needed a bit more investigation and medical/nursing care than can be put in at home. As stated, mainly this was for older patients, but we had on the spot testing for DVTs (blood clots in the leg), inflammatory markers (potential signs of infection), Xrays, etc so operated a Clinical Decision Unit - patients would come here, get some testing and may be sent home, admitted to the Community Hospital or transferred to the main hospital. we were, however, fairly rural, so for bigger cities, this function is likely to be performed by a normal hospital. We stopped frail elderly patients filling up beds/chairs/waiting areas whilst awaiting results and transfer home or to the ward. There were also dedicated OTs and Physios and regular visits from SaLT.
As said, though, inevitably beds were full of step down patients, usually elderly, often post fracture/surgery who needed a bit of convalesence with nurses available 24hrs. We did a medical ward round on most of them daily (maybe not those just waiting for a Care Home placement) and a Consultant came out once every 2 weeks to advise about difficult cases, etc. As the beds were nearly always full, a lot of the step up care is now dealt with by Community Multidisciplinary Teams who are (supposed to be) able to be mobilised at short notice to care for people in their own homes. We have a Community Rehab Team who consist of Nurses, OTs, Physios and HCAs/carers who can visit a patient up to 4 times a day in their own homes to try and avoid hospital admission. Clinical care remains with their own GP in this scenario and we would meet with members of this team a minimum of weekly to reassess needs (though is often daily if things are changing rapidly). The advantage to this is the GP often knows the patient and their family/community very well, so is more likely to recognise changes and instigate a patient-centred care plan as they have prior knowledge of a patient's views.
I would disagree that the support at Community Hospitals needs to be the same as at a standard hospital, otherwise there is no point having both! Patients are carefully selected as fit for transfer to a Community Hospital or not needing secondary care if step up - remember, it is frequently the GP who decides to admit a patient, so makes that call every day. They need good medical care, nursing support, mainly Physio and OT input and this happens daily in the community anyway, just these patients need it 24/7 rather than intermittently, but they need it at a level consistent with Primary Care doctors and nurses, not those from a hospital. We have had acute medics really struggle in a Community setting as they have tried to "cure" everybody there, when that is rarely the demographic of patient we are talking about! Secondary care input, if provided, is best done by somebody with community older adults experience.
As also stated, one of those hospitals had a nurse-led MIU (with GP input if needed). The other did too, but it shut down due to changes in funding several years ago. They also both provide out patient services for clinics by Consultants (and at 1, we run combined clincis with Paeds consultants and the pt's GP so knowledge and learning can be shared) and they have out patient Physio services on site as well as other visiting clinics (podiatry, retinal screening, audiology, dietitian, etc).
Community Hospitals' existence has been under threat for years now, as they are considered unnecessary by those that know nothing about how the NHS works, but patients generally love them and prefer the local and often more personalised approach they get at them. Most of my patients plead to be admitted to our local Community Hospital if they have to go in and explaining they cannot do their appendicectomy or repair a fractured hip is a common discussion!