Lead Acid batteries are usually charged at a constant voltage but generally there is no need to control the current. The current during the charge is typically controlled by the charge state of the battery. It starts off high and falls as the battery charges. So there is a form of regulation.
Having said that.. It is normal for the charger to have a current limit which comes into play at the start of the charge cycle when the battery is flat. The current at the end of the charge is controlled by the voltage which needs to be quite accurate. If the voltage and consequently the current is too high at the end then gassing occurs for the reason hvogt gives.
With certain conditions Lead Acid batteries can actually be left on charge permanantly. This is called float charging and is usually done at a slightly lower voltage than normal to ensure the final current is quite low and minimise gassing that would dry out the cell. Typically on float the current will fall to close to zero - just balancing the self discharge losses. In fact wet Lead acid batteries seem to last longer if kept permanantly on a correctly set up float charger. They typically hate being deep discharged, allowed to self discharge or allowed to stand flat for even quite short periods. This applies even to wet cells designed for deep discharge.
I've flown electric powered model planes for 20 years and in my first few seasons I managed to kill three lead acid batteries a year by allowing them to stand party discharged for too long. Now as soon as I get back from a days flying the battery is put on float charge immediatly and is left like that 24/7. Now the battery lasts several years.
I highly recommend using a charger with a float setting if you have a battery powered lawn mower that uses wet lead acid batteries. Keep it on float over winter. Sometimes the float voltage can be adjusted to match that recommended by the battery manufacturer.