I see two possible meanings here.
1. As MH said - they may not have enough members of staff.
Thing's are a bit tight financially, at the moment - I only just about have enough money for my needs at the moment.
2. Tight can be used in the sense of strict control.
We're tight on staffing at the moment - we are watching very closely to make sure that everything is as it should be - number of staff, behaviour of staff members etc .
Some places are tight on security = they have strict security measures in place which are adhered to.