it’s not your standard email response. It’s [your name’s] out-of-office because [he/she/they] are on holidays and will respond when they return.
If the person keeps you on the message thread in which they then contact someone else (ideally, the person you stated in your out of office message as the one who’s handling urgent issues in your absence), you are literally kept in the loop and can see for yourself that all was handled when you get back. Of course, this works even better if you check the most recent items in your inbox first when you get back.

One may also ask, how do you write a holiday notice? Holiday Notice Letter Writing Tips Specify the duration of leave and name of the colleague who will handle your work responsibilities in your absence. Furnish contact details for any urgent requirement and mention your date of joining back work. I will be out of the office starting (Starting Date) through (End Date) returning(Date of Return). If you need immediate assistance during my absence, please contact (Contacts Name) at (Contacts Email Address). Otherwise I will respond to your emails as soon as possible upon my return. Include the date of your return. Let people know of any other individuals they can contact for help in your absence. If people typically contact you for a certain type of information or approval, let readers know how they can find the information or get the approval in your absence. How do you write a professional out of office message?
If instead you ask your co-workers to cc or bcc on replies then you will know which have been dealt with. (I think for internal mails it’s more reasonable to ask that if the original person contacts someone else, they cc you so you know who is dealing – and in smaller organisations where people know you personally you could also send a mail round the day before you leave to say you’re going to be out and to ask that any enquiries are directed to [name]in your absence, to try to avoid them coming into your inbox in the first place.