Thank you for your question and welcome to Just Answer.
In an ideal situation, the success rate of an implant is well above 95%. In these 5% of implants that do fail, after the appropriate amount of healing time, the success rate of the second implant in the same site is again 95%. Therefore the odds that you would have two implants fail in the same surgical site are quite low which makes me suspect that there other factors at play here that will compromise the success of the implant integration. I will outline some common reasons that implants fail, and if you don't think that any of these pertain to you, then I would say try for the third placement, especially as your periodontist is not charging you for the replacement (which he shouldn't anyway). Although a bridge is an acceptable solution to replacing a missing tooth, it does involve preparing two perfectly good teeth. Therefore if you can get an implant to successfully integrate and then have a single crown placed over that implant, you have a much more conservative solution to replacing your missing tooth.
The most common situation that causes implants to fail is smoking. Successful implant integration requires that smokers stop smoking a few months prior to the surgery and to ideally not smoke for at least three months after that. Smoking severely constricts the blood flow to the oral tissues and prevents the nutrients necessary for new bone growth to reach the surgical site.
For the same reason, patients with diabetes are at risk for implant failure. Their peripheral circulation is compromised and they have a longer healing time for any surgical procedure.
For an implant to be successful, there must be enough bone in the jaw to support the implant, typically at least 6 millimeters. Any less than that and a bone graft may be needed prior to the implant placement.
There is new information about intravenous and oral bisphosphonates (taken for certain forms of breast cancer and osteoporosis, respectively) which may put patients at a higher risk of developing a delayed healing syndrome called osteonecrosis. Implants are contraindicated for some patients who take intravenous bisphosphonates.
I would recommend, if it wasn't done during your first two surgeries, that you be placed on antibiotics before and after the implant placement to prevent any possibility that an infection sets in.
Finally, sterile technique is a must during the surgery. I'm sure your periodontist is not amiss in this category, but any contamination of the implant, which comes in a sterilized packet before it is placed in the mouth, will cause an immediate failure.
I hope this answers your question. If you would like to discuss this further or have any additional questions, please reply to my answer and I will get right back to you.
Sincerely, XXXXX XXXXX DDS
Did you by any chance have a temporary tooth attached to the implant imediately after placement, or was there just a small healing cap placed that did not touch any other teeth?