I would expect the firm to publish its accounts to all partners, including fixed share/salaried, including partner distributions and current accounts. Why don't you ask them for the previous years accounts, particularly if you are being asked to invest money in the business.
And as one poster has said, there is likely to be a lot more required of you for the extra wedge. Law, like other sectors, is now much more demanding generally and less well remunerated than previously. It is also risky in a way that it didn't used to be and there have been a number of firms going into administration over the last few years - most recently, Linder Myers. The US firms and the MC firms are still doing fine but the mid tier firms and some of the smaller ones are struggling, as you will be aware. If you go onto Legal Week, the Lawyer or Roll on Friday, there is a lot of information which may be useful, including details of average profits earned per equity partner (PEP).
However, the PEP info does have to be treated with a bit of caution; they are a bit like OFSTEAD tables, in that they can be manipulated. After I left my previous firm and received my last share of profits, I concluded that it had been an especially bad year, as the profits were down about 20-25% against budget and I didn't even receive my return on capital invested in the business. I later queried the figures after reading in the PEP league tables that the figures were considerably higher, but was advised that this was for end of year financial adjustments after the accounts had been finalised....! Also, members are now required to invest £500k in the business, which amounts to a fair amount of risk, especially at an average return of £200-£275k for a London equity member in a large national firm, before payment of interest on loans, expenses and higher rate tax.
So OP, congrats on your progress, but you are right to query the financial position and to go into it with your eyes open!