In most cases you will both need to employ separate divorce solicitors to advise you and act for you in negotiations and possibly at Court, so you will both incur costs.
The person initiating the divorce petition (the ‘petitioner’) will incur additional Court fees on top of the fees for their solicitor preparing the case. The person being petitioned against (the ‘respondent’) will incur the costs of their divorce solicitor preparing a reply to the petition.
The petitioner may claim some or all of his/her divorce costs from the respondent. Costs orders are typically made on petitions citing adultery or unreasonable behaviour.
In cases where an amicable agreement can be reached, it is often possible for the costs of the divorce to be shared equally between the parties and for no orders to be made. This is distinct from the costs of resolving the financial/property issues and any issues concerning the children, where the general rule is that each party is responsible for payment of their own legal fees.