For many the bond you have with your love is very deep so grieving is only natural, how you deal with that grief is a personal thing. In my case it took me a while as I was their for his birth, so to say we spent a lot of time together for the next 14 years was an understatement. And everywhere I turned he just wasn't there anymore beside my side, wanting to be involved with what I was doing, even if that just meant laying on a dog bed nearby.
There is something I would like to comment on though:
If you have a second animal then it is also very important to recognise that it is not just you that will be grieving. Most of the time the second animal will be dealing with the loss also. So if your in that position openly showing your distress for the loss in the beginning in front of the other animal is perfectly fine, it is a very emotional thing for the two of you. Following that spending a lot of time with them will help both of you to comfort each other, and if nothing else give you a reason to get out of bed in the morning, because they still need you. This will also let you gauge how they are dealing with the passing. When they seem to have accepted their friend is gone and you feel you're ready, that is the time to consider bringing someone new into the pack/herd.
I am saying this because bringing a new animal into the fold can be stressful for some animals at the best of times, but if they are dealing with the loss of someone that was close to them, can push them to the breaking point; resulting in them uncharacteristically lashing out. Particularly if you bring a young animal into the mix who wants to play and jump on the older animal going through the grieving process, and thus has zero patients for the shenanigans of the new intruder into the house/barn/paddock.
Generally it's all about getting the timing right, too early and you can create issues between animals that wouldn't occur otherwise and getting their relationship off on the wrong paw/hoof. But in time the introduction of a new friend or playmate can help put life back into both you and your existing animal. In the case of dogs it is also important in the beginning to give the existing animal top dog status and the pup subordinate status. eg: the order you give food in if putting out bowls sequentially rather than all at once. In time the two dogs will sort out who's the boss and who's the follower, but giving the existing dog top dog status will help with the transition of bringing in a new face run more smoothly.
So do what your comfortable with, but keep the emotions of others in mind if they are also going to have to live with your decision to bring someone new into your lives.
