I can confirm that these offers of visits are genuine. It is not just Pension credit, which is means tested, but ALL other non means tested benefits. eg Attendance Allowance, Carers Allowances, Housing Benefit and Council Tax Benefit that they will look into. They will also advise you of any local companies/agencies offering assistance/help to your age group.
As for savings anything under �6000 is not taken into the assessment. If you are over 65 then for every �1000 over �6000 it is considered to be �2 per week income. This would be then be added to your joint retirement pensions and any occupational pensions you may have ( and any other money coming in) to establish your weekly income. You can actually have quite a bit of savings, a good few thousand, and still qualify for Pension Credit.
I can assure you they aren't snoopers they are just trying to get you more income or assistance.
You can always refuse to be visited but you would never know if you are missing out on extra money.