Here are some extensions methods that handle string and binary serialization of Point3DCollection bags. As I said in my comment, I don't think there is a best way of doing this in all cases, so you might want to try both. Also note they're using Stream parameter as input so you can chain these with calls to GZipStream of DeflateStream.

public static class Point3DExtensions
{
    public static void StringSerialize(this ConcurrentBag<Point3DCollection> bag, Stream stream)
    {
        if (bag == null)
            throw new ArgumentNullException("bag");

        if (stream == null)
            throw new ArgumentNullException("stream");

        StreamWriter writer = new StreamWriter(stream);
        Point3DCollectionConverter converter = new Point3DCollectionConverter();
        foreach (Point3DCollection coll in bag)
        {
            // we need to use the english locale as the converter needs that for parsing...
            string line = (string)converter.ConvertTo(null, CultureInfo.GetCultureInfo("en-US"), coll, typeof(string));
            writer.WriteLine(line);
        }
        writer.Flush();
    }

    public static void StringDeserialize(this ConcurrentBag<Point3DCollection> bag, Stream stream)
    {
        if (bag == null)
            throw new ArgumentNullException("bag");

        if (stream == null)
            throw new ArgumentNullException("stream");

        StreamReader reader = new StreamReader(stream);
        Point3DCollectionConverter converter = new Point3DCollectionConverter();
        do
        {
            string line = reader.ReadLine();
            if (line == null)
                break;

            bag.Add((Point3DCollection)converter.ConvertFrom(line));

            // NOTE: could also use this:
            //bag.Add(Point3DCollection.Parse(line));
        }
        while (true);
    }

    public static void BinarySerialize(this ConcurrentBag<Point3DCollection> bag, Stream stream)
    {
        if (bag == null)
            throw new ArgumentNullException("bag");

        if (stream == null)
            throw new ArgumentNullException("stream");

        BinaryWriter writer = new BinaryWriter(stream);
        writer.Write(bag.Count);
        foreach (Point3DCollection coll in bag)
        {
            writer.Write(coll.Count);
            foreach (Point3D point in coll)
            {
                writer.Write(point.X);
                writer.Write(point.Y);
                writer.Write(point.Z);
            }
        }
        writer.Flush();
    }

    public static void BinaryDeserialize(this ConcurrentBag<Point3DCollection> bag, Stream stream)
    {
        if (bag == null)
            throw new ArgumentNullException("bag");

        if (stream == null)
            throw new ArgumentNullException("stream");

        BinaryReader reader = new BinaryReader(stream);
        int count = reader.ReadInt32();
        for (int i = 0; i < count; i++)
        {
            int pointCount = reader.ReadInt32();
            Point3DCollection coll = new Point3DCollection(pointCount);
            for (int j = 0; j < pointCount; j++)
            {
                coll.Add(new Point3D(reader.ReadDouble(), reader.ReadDouble(), reader.ReadDouble()));
            }
            bag.Add(coll);
        }
    }
}

And a little console app test program to play with:

    static void Main(string[] args)
    {
        Random rand = new Random(Environment.TickCount);
        ConcurrentBag<Point3DCollection> bag = new ConcurrentBag<Point3DCollection>();
        for (int i = 0; i < 100; i++)
        {
            Point3DCollection coll = new Point3DCollection();
            bag.Add(coll);

            for (int j = rand.Next(10); j < rand.Next(100); j++)
            {
                Point3D point = new Point3D(rand.NextDouble(), rand.NextDouble(), rand.NextDouble());
                coll.Add(point);
            }
        }

        using (FileStream stream = new FileStream("test.bin", FileMode.Create))
        {
            bag.StringSerialize(stream); // or Binary
        }

        ConcurrentBag<Point3DCollection> newbag = new ConcurrentBag<Point3DCollection>();
        using (FileStream stream = new FileStream("test.bin", FileMode.Open))
        {
            newbag.StringDeserialize(stream); // or Binary
            foreach (Point3DCollection coll in newbag)
            {
                foreach (Point3D point in coll)
                {
                    Console.WriteLine(point);
                }
                Console.WriteLine();
            }
        }
    }
}