If a party is requested to sign an assent, there are documents that will be submitted to the court which explain the proposed action. The first opportunity to assent is to approve the will and appoint the executor. The subsequent assents could be all sorts of things. Nobody should sign an assent unless they have copies of the documents relating to the issue and understand the contents. The assent will identify the document that is submitted to the judge. An assent says to the court that you approve the proposed action of the executor. It also says to the court that you don’t need further notice of the proposed action.
The following is an example of how this works. When the probate is filed, the petitioner files a copy of the will with a petition to probate, a death certificate, and a bond. If you sign the assent, you probably won’t get further notice of the petition to probate. If you don’t sign the assent, you will be “served” (probably by mail but possibly by a sheriff) with a court document called an order of notice that tells you the petition and will were filed and that you are an interested party and have the right to object to the petition. It will also tell you the date by which you must file your objection. If you don’t file the objection by that date, you will not have the right to contest the will or appointment.