Intermittent image upload and database entries! Not sure if it's my PHP or SQL. Help!

I can't offer a solution, but it looks to me like you may be being overly optimistic about assuming your MySQL queries will succeed every time. I'm not too familiar with PHP or mysqli any more, but from a quick search it looks as though if an error arises in processing the statement, it will return false rather than generating any error, and you never check for this. I would suggest checking the results of the executions of your insert statements for true/false, and if false log the error details. You could have failing inserts into your images tables but be glossing over them at the moment.

Also, as a slight digression, is there a particular reason you went for separate public and private image tables rather than just having one table with a flag column (e.g. enum) or columns (boolean for each of public/private) on it? It seems like a significant duplication of both data and code to have to insert/update across both tables.

I guess it might be quicker (depending on how you do things) in that the data being filtered for each query is already a subset, but it may well mean having to do two queries and aggregation in memory on the application side (often less efficient), or otherwise unnecessary usage of UNION statements to generate resultsets from both. If you never want to display the data from both tables together then this might not be a concern, but ultimately I wouldn't imagine the performance hit from a consolidated table being anything that couldn't be overcome by proper design and indexing on the filtering columns.

And now that you mention escaping the captions, if that has improved things then it's actually very possible that you could have had some issues with (non-malicious, and accidental) SQL injection inadvertently rendering your statements invalid - again checking for success around the DB calls might have revealed that, even if you just tried to log the failing statement if false came back. This is where prepared statements are often invaluable, as they do a lot of this boilerplate for you.

As has already been suggested I think you should look at the architecture of your application.

Instead of maintaining two database tables (Public and Private) you should only need a single table - keep things a simple as possible so they are easy to manage and troubleshoot.

You can add a field to indicate the status of image.
Value: 1 -> Private
Value: 2 -> Public
Value: 3 -> Private and Public

Users should not be posting data directly to your database for two reasons.
#1 : Security - the last thing you want to happen is the data to be poisoned.
#2 : Scalability - how many concurrent uploads will the system be able to handle?

Instead implement a queue. Upload the image to the server or Amazon S3 and attach a reference to it in a data structure along with the other posted data. A server side service should iterate through the queue and process the scrubbed data. This process will provide the database with an additional layer of protection and also make your application more scalable. The queue structure should also resolve the issue you are encountering with data being lost.

Ideally the data should be inserted using prepared statements, unfortunately for you MySQLi doesn't support prepared statements but does support parameter binding.

Your current SQL statement:
[PHP]$publicSQLQuery = "INSERT INTO publicImages (id, image, caption, lat, lon, address, time, sender) VALUES ('', '$imageURL', '$caption', '$lat', '$lon', '$address', '$timeAndDate','$uploader')";[/PHP]

A more secure MySQLi method using parameter binding:

[PHP]$publicInsert = $mysqli->prepare(
"INSERT INTO publicImages (image, caption, lat, lon, address, time, sender)
VALUES (?, ?, ?, ?, ?, ?, ?)"
);
$publicInsert->bind_param($imageURL, $caption, $lat, $lon, $address, $timeAndDate, $uploader);
$publicInsert->execute();[/PHP]

Alternatively using PDO instead of MySQLi to connect to the database gives you the option of using a prepared statement:

[PHP]$params = array(
':image' => $imageURL,
':caption' => $caption,
':lat' => $lat,
':lon' => $lon,
':address' => $address,
':time' => $timeAndDate,
':sender' => $uploader
);

$pdo->prepare(
"INSERT INTO publicImages (image, caption, lat, lon, address, time, sender)
VALUES (:image, :caption, :lat, :lon, :address, :time, :sender)"
);

$pdo->execute($params);[/PHP]

In your second post you mention fixing bugs by escaping input strings, the PDO::quote() function places quotes around the input string if required and escapes special characters within the string, it uses a quote style that is appropriate to the underlying database driver. PDO takes the headache out of making the decision whether to escape the string or not and gives you a better workflow.

PDO also gives you the option of switching to an alternative database without having to rewrite chunks of the application. MySQLi only supports MySQL, PDO gives you the option of switching out MySQL for almost any alternative. For example you might decide that PostgreSQL is a better option for your application because PostgreSQL + PostGIS is a fully featured spatial database, MySQL was late to the spatial database party and really doesn't compare.