You are a trustee of one of these trusts. If you do not already have a copy of the trust deed then you must get one - if the trust was drawn up by the solicitor he/she must have the document. You have every right to see it & it must be given to you on request.
Similarly, any of the trustees of the other trust can request a copy of that deed.
It is possible that you will find the deeds do not specify who has to sign documents. If they do not, then there are rules laid down in Trust law which will govern this, but you are unlikely to be able to clarify it definitively without a solicitor (one experienced in trust law) being involved. The same applies to the question of keeping records & producing accounts, although it seems entirely reasonable for all the trustees to be provided with accounts showing the position of the trust in sufficient detail for them to be able to check & satisfy themselves as to what has been done. I believe some sort of annual tax return has to be done for most (if not all) trusts & accounts must be produced for this. All trustees are liable for the correct use of the trust funds & consequently all have a right (& would be very unwise to forgo the right) to be fully aware of what is being done with the trust funds.
I suggest you (& the other non-solicitor trustees if you all have the same concerns) make an appointment to see the solicitor & raise all your queries with him/her. There may well be a sensible explanation. If you cannot get satisfaction you may have to take other action. This could involve the cost of another solicitor or making a formal complaint using the Law Society procedure (see their website) against the trustee solicitor.